{"id":19867,"date":"2022-02-23T01:31:47","date_gmt":"2022-02-23T01:31:47","guid":{"rendered":"https:\/\/onlinemathtutor.co\/?page_id=19867"},"modified":"2022-05-19T20:57:33","modified_gmt":"2022-05-19T20:57:33","slug":"integration-practice-problems-application-rotational-volumes","status":"publish","type":"page","link":"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/integration\/problem-set-integration\/integration-practice-problems-application-rotational-volumes\/","title":{"rendered":"Integration Practice Problems &#8211; [Application: Rotational Volumes]"},"content":{"rendered":"\t\t<div data-elementor-type=\"wp-page\" data-elementor-id=\"19867\" class=\"elementor elementor-19867\" data-elementor-post-type=\"page\">\n\t\t\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-122add1 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"122add1\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-f69b248\" data-id=\"f69b248\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-fba6159 elementor-widget elementor-widget-heading\" data-id=\"fba6159\" data-element_type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<h1 class=\"elementor-heading-title elementor-size-default\">Practice Problems: Rotational Volumes<\/h1>\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-9ac37bb elementor-nav-menu__align-center elementor-nav-menu--stretch elementor-nav-menu__text-align-center animated-fast elementor-nav-menu--dropdown-tablet elementor-nav-menu--toggle elementor-nav-menu--burger elementor-widget elementor-widget-nav-menu\" data-id=\"9ac37bb\" data-element_type=\"widget\" data-settings=\"{&quot;submenu_icon&quot;:{&quot;value&quot;:&quot;&lt;i class=\\&quot;fas fa-pencil-alt\\&quot;&gt;&lt;\\\/i&gt;&quot;,&quot;library&quot;:&quot;fa-solid&quot;},&quot;full_width&quot;:&quot;stretch&quot;,&quot;_animation&quot;:&quot;none&quot;,&quot;layout&quot;:&quot;horizontal&quot;,&quot;toggle&quot;:&quot;burger&quot;}\" data-widget_type=\"nav-menu.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t<nav aria-label=\"Menu\" class=\"elementor-nav-menu--main elementor-nav-menu__container elementor-nav-menu--layout-horizontal e--pointer-double-line e--animation-fast\">\n\t\t\t\t<ul id=\"menu-1-9ac37bb\" class=\"elementor-nav-menu\"><li class=\"menu-item menu-item-type-custom menu-item-object-custom menu-item-19677\"><a href=\"https:\/\/onlinemathtutor.co\/index.php\/problem-set-integration\/\" class=\"elementor-item\"><img decoding=\"async\" src=\"https:\/\/onlinemathtutor.co\/wp-content\/uploads\/2021\/06\/Home-Icon-Blue.png\"  width=\"20\"><\/a><\/li>\n<li class=\"menu-item menu-item-type-custom menu-item-object-custom menu-item-has-children menu-item-19577\"><a href=\"#\" class=\"elementor-item elementor-item-anchor\">Integration Techniques<\/a>\n<ul class=\"sub-menu elementor-nav-menu--dropdown\">\n\t<li class=\"menu-item menu-item-type-custom menu-item-object-custom menu-item-has-children menu-item-19777\"><a href=\"#\" class=\"elementor-sub-item elementor-item-anchor\">U-Substitution<\/a>\n\t<ul class=\"sub-menu elementor-nav-menu--dropdown\">\n\t\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-19576\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/integration\/problem-set-integration\/integration-practice-problems-usub\/\" class=\"elementor-sub-item\">Problem Set<\/a><\/li>\n\t\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-19776\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/integration\/problem-set-integration\/virtual-lessons-integration-usub\/\" class=\"elementor-sub-item\">Virtual Lessons<\/a><\/li>\n\t<\/ul>\n<\/li>\n\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-has-children menu-item-19694\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/integration\/problem-set-integration\/integration-practice-problems-ibp\/\" class=\"elementor-sub-item\">Integration By Parts<\/a>\n\t<ul class=\"sub-menu elementor-nav-menu--dropdown\">\n\t\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-19898\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/integration\/problem-set-integration\/integration-practice-problems-ibp\/\" class=\"elementor-sub-item\">Problem Set<\/a><\/li>\n\t\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-19956\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/integration\/problem-set-integration\/virtual-lessons-integration-ibp\/\" class=\"elementor-sub-item\">Virtual Lessons<\/a><\/li>\n\t<\/ul>\n<\/li>\n\t<li class=\"menu-item menu-item-type-custom menu-item-object-custom menu-item-has-children menu-item-19885\"><a href=\"#\" class=\"elementor-sub-item elementor-item-anchor\">Trigonometric Integrals<\/a>\n\t<ul class=\"sub-menu elementor-nav-menu--dropdown\">\n\t\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-19925\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/integration\/problem-set-integration\/integration-practice-problems-trigonometric-integrals\/\" class=\"elementor-sub-item\">Problem Set<\/a><\/li>\n\t\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-19954\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/integration\/problem-set-integration\/virtual-lessons-integration-trigonometric-integrals\/\" class=\"elementor-sub-item\">Virtual Lessons<\/a><\/li>\n\t<\/ul>\n<\/li>\n\t<li class=\"menu-item menu-item-type-custom menu-item-object-custom menu-item-has-children menu-item-19886\"><a href=\"#\" class=\"elementor-sub-item elementor-item-anchor\">Trigonometric Substitution<\/a>\n\t<ul class=\"sub-menu elementor-nav-menu--dropdown\">\n\t\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-19926\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/integration\/problem-set-integration\/integration-practice-problems-trigonometric-substitution\/\" class=\"elementor-sub-item\">Problem Set<\/a><\/li>\n\t\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-19953\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/integration\/problem-set-integration\/virtual-lessons-integration-trigonometric-substitution\/\" class=\"elementor-sub-item\">Virtual Lessons<\/a><\/li>\n\t<\/ul>\n<\/li>\n\t<li class=\"menu-item menu-item-type-custom menu-item-object-custom menu-item-has-children menu-item-19884\"><a href=\"#\" class=\"elementor-sub-item elementor-item-anchor\">Partial Fractions<\/a>\n\t<ul class=\"sub-menu elementor-nav-menu--dropdown\">\n\t\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-19924\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/integration\/problem-set-integration\/integration-practice-problems-partial-fractions\/\" class=\"elementor-sub-item\">Problem Set<\/a><\/li>\n\t\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-19955\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/integration\/problem-set-integration\/virtual-lessons-integration-partialfractions\/\" class=\"elementor-sub-item\">Virtual Lessons<\/a><\/li>\n\t<\/ul>\n<\/li>\n\t<li class=\"menu-item menu-item-type-custom menu-item-object-custom menu-item-has-children menu-item-19893\"><a href=\"#\" class=\"elementor-sub-item elementor-item-anchor\">Improper Integrals<\/a>\n\t<ul class=\"sub-menu elementor-nav-menu--dropdown\">\n\t\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-19927\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/integration\/problem-set-integration\/integration-practice-problems-improper-integrals\/\" class=\"elementor-sub-item\">Problem Set<\/a><\/li>\n\t\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-19966\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/integration\/problem-set-integration\/virtual-lessons-integration-improper-integrals\/\" class=\"elementor-sub-item\">Virtual Lessons<\/a><\/li>\n\t<\/ul>\n<\/li>\n<\/ul>\n<\/li>\n<li class=\"menu-item menu-item-type-custom menu-item-object-custom menu-item-has-children menu-item-19578\"><a href=\"#\" class=\"elementor-item elementor-item-anchor\">Integration Applications<\/a>\n<ul class=\"sub-menu elementor-nav-menu--dropdown\">\n\t<li class=\"menu-item menu-item-type-custom menu-item-object-custom menu-item-has-children menu-item-19888\"><a href=\"#\" class=\"elementor-sub-item elementor-item-anchor\">Area<\/a>\n\t<ul class=\"sub-menu elementor-nav-menu--dropdown\">\n\t\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-19928\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/integration\/problem-set-integration\/integration-practice-problems-application-area\/\" class=\"elementor-sub-item\">Problem Set<\/a><\/li>\n\t\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-20136\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/integration\/problem-set-integration\/virtual-lessons-integration-area\/\" class=\"elementor-sub-item\">Virtual Lessons<\/a><\/li>\n\t<\/ul>\n<\/li>\n\t<li class=\"menu-item menu-item-type-custom menu-item-object-custom menu-item-has-children menu-item-19889\"><a href=\"#\" class=\"elementor-sub-item elementor-item-anchor\">Volumes [Nonrotational]<\/a>\n\t<ul class=\"sub-menu elementor-nav-menu--dropdown\">\n\t\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-19929\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/integration\/problem-set-integration\/integration-practice-problems-application-volume\/\" class=\"elementor-sub-item\">Problem Set<\/a><\/li>\n\t\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-20582\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/integration\/problem-set-integration\/virtual-lessons-integration-trigonometric-substitution\/\" class=\"elementor-sub-item\">Virtual Lessons<\/a><\/li>\n\t<\/ul>\n<\/li>\n\t<li class=\"menu-item menu-item-type-custom menu-item-object-custom menu-item-has-children menu-item-19890\"><a href=\"#\" class=\"elementor-sub-item elementor-item-anchor\">Volume [Rotational]<\/a>\n\t<ul class=\"sub-menu elementor-nav-menu--dropdown\">\n\t\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-19931\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/integration\/problem-set-integration\/integration-practice-problems-application-rotational-volumes\/\" class=\"elementor-sub-item\">Problem Set<\/a><\/li>\n\t\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-20160\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/integration\/problem-set-integration\/virtual-lessons-integration-rotational-volume\/\" class=\"elementor-sub-item\">Virtual Lessons<\/a><\/li>\n\t<\/ul>\n<\/li>\n\t<li class=\"menu-item menu-item-type-custom menu-item-object-custom menu-item-has-children menu-item-19891\"><a href=\"#\" class=\"elementor-sub-item elementor-item-anchor\">Surface Area<\/a>\n\t<ul class=\"sub-menu elementor-nav-menu--dropdown\">\n\t\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-19930\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/integration\/problem-set-integration\/integration-practice-problems-application-surface-area\/\" class=\"elementor-sub-item\">Problem Set<\/a><\/li>\n\t\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-20165\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/integration\/problem-set-integration\/virtual-lessons-integration-surface-area\/\" class=\"elementor-sub-item\">Virtual Lessons<\/a><\/li>\n\t<\/ul>\n<\/li>\n\t<li class=\"menu-item menu-item-type-custom menu-item-object-custom menu-item-has-children menu-item-19892\"><a href=\"#\" class=\"elementor-sub-item elementor-item-anchor\">Arc Length<\/a>\n\t<ul class=\"sub-menu elementor-nav-menu--dropdown\">\n\t\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-20177\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/integration\/problem-set-integration\/integration-practice-problems-arc-length\/\" class=\"elementor-sub-item\">Problem Set<\/a><\/li>\n\t\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-20176\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/integration\/problem-set-integration\/virtual-lessons-integration-arc-length\/\" class=\"elementor-sub-item\">Virtual Lessons<\/a><\/li>\n\t<\/ul>\n<\/li>\n<\/ul>\n<\/li>\n<li class=\"menu-item menu-item-type-custom menu-item-object-custom menu-item-has-children menu-item-19579\"><a href=\"#\" class=\"elementor-item elementor-item-anchor\">Numerical Integration<\/a>\n<ul class=\"sub-menu elementor-nav-menu--dropdown\">\n\t<li class=\"menu-item menu-item-type-custom menu-item-object-custom menu-item-has-children menu-item-19894\"><a href=\"#\" class=\"elementor-sub-item elementor-item-anchor\">Reimann Sums<\/a>\n\t<ul class=\"sub-menu elementor-nav-menu--dropdown\">\n\t\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-20110\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/integration\/problem-set-integration\/integration-practice-problems-riemann-sums\/\" class=\"elementor-sub-item\">Problem Set<\/a><\/li>\n\t\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-20128\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/integration\/problem-set-integration\/virtual-lessons-integration-riemann-sum\/\" class=\"elementor-sub-item\">Virtual Lessons<\/a><\/li>\n\t<\/ul>\n<\/li>\n\t<li class=\"menu-item menu-item-type-custom menu-item-object-custom menu-item-has-children menu-item-19895\"><a href=\"#\" class=\"elementor-sub-item elementor-item-anchor\">Trapezoidal Rule<\/a>\n\t<ul class=\"sub-menu elementor-nav-menu--dropdown\">\n\t\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-20109\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/integration\/problem-set-integration\/integration-practice-problems-trapezoidal\/\" class=\"elementor-sub-item\">Problem Set<\/a><\/li>\n\t\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-20138\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/integration\/problem-set-integration\/virtual-lessons-integration-trapezoidal\/\" class=\"elementor-sub-item\">Virtual Lessons<\/a><\/li>\n\t<\/ul>\n<\/li>\n\t<li class=\"menu-item menu-item-type-custom menu-item-object-custom menu-item-has-children menu-item-19896\"><a href=\"#\" class=\"elementor-sub-item elementor-item-anchor\">Simpson&#8217;s Rule<\/a>\n\t<ul class=\"sub-menu elementor-nav-menu--dropdown\">\n\t\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-20108\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/integration\/problem-set-integration\/integration-practice-problems-simpsons\/\" class=\"elementor-sub-item\">Problem Set<\/a><\/li>\n\t\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-20137\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/integration\/problem-set-integration\/virtual-lessons-integration-simpsons\/\" class=\"elementor-sub-item\">Virtual Lessons<\/a><\/li>\n\t<\/ul>\n<\/li>\n<\/ul>\n<\/li>\n<\/ul>\t\t\t<\/nav>\n\t\t\t\t\t<div class=\"elementor-menu-toggle\" role=\"button\" tabindex=\"0\" aria-label=\"Menu Toggle\" aria-expanded=\"false\">\n\t\t\t<i aria-hidden=\"true\" role=\"presentation\" class=\"elementor-menu-toggle__icon--open eicon-menu-bar\"><\/i><i aria-hidden=\"true\" role=\"presentation\" class=\"elementor-menu-toggle__icon--close eicon-close\"><\/i>\t\t<\/div>\n\t\t\t\t\t<nav class=\"elementor-nav-menu--dropdown elementor-nav-menu__container\" aria-hidden=\"true\">\n\t\t\t\t<ul id=\"menu-2-9ac37bb\" class=\"elementor-nav-menu\"><li class=\"menu-item menu-item-type-custom menu-item-object-custom menu-item-19677\"><a href=\"https:\/\/onlinemathtutor.co\/index.php\/problem-set-integration\/\" class=\"elementor-item\" tabindex=\"-1\"><img decoding=\"async\" src=\"https:\/\/onlinemathtutor.co\/wp-content\/uploads\/2021\/06\/Home-Icon-Blue.png\"  width=\"20\"><\/a><\/li>\n<li class=\"menu-item menu-item-type-custom menu-item-object-custom menu-item-has-children menu-item-19577\"><a href=\"#\" class=\"elementor-item elementor-item-anchor\" tabindex=\"-1\">Integration Techniques<\/a>\n<ul class=\"sub-menu elementor-nav-menu--dropdown\">\n\t<li class=\"menu-item menu-item-type-custom menu-item-object-custom menu-item-has-children menu-item-19777\"><a href=\"#\" class=\"elementor-sub-item elementor-item-anchor\" tabindex=\"-1\">U-Substitution<\/a>\n\t<ul class=\"sub-menu elementor-nav-menu--dropdown\">\n\t\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-19576\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/integration\/problem-set-integration\/integration-practice-problems-usub\/\" class=\"elementor-sub-item\" tabindex=\"-1\">Problem Set<\/a><\/li>\n\t\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-19776\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/integration\/problem-set-integration\/virtual-lessons-integration-usub\/\" class=\"elementor-sub-item\" tabindex=\"-1\">Virtual Lessons<\/a><\/li>\n\t<\/ul>\n<\/li>\n\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-has-children menu-item-19694\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/integration\/problem-set-integration\/integration-practice-problems-ibp\/\" class=\"elementor-sub-item\" tabindex=\"-1\">Integration By Parts<\/a>\n\t<ul class=\"sub-menu elementor-nav-menu--dropdown\">\n\t\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-19898\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/integration\/problem-set-integration\/integration-practice-problems-ibp\/\" class=\"elementor-sub-item\" tabindex=\"-1\">Problem Set<\/a><\/li>\n\t\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-19956\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/integration\/problem-set-integration\/virtual-lessons-integration-ibp\/\" class=\"elementor-sub-item\" tabindex=\"-1\">Virtual Lessons<\/a><\/li>\n\t<\/ul>\n<\/li>\n\t<li class=\"menu-item menu-item-type-custom menu-item-object-custom menu-item-has-children menu-item-19885\"><a href=\"#\" class=\"elementor-sub-item elementor-item-anchor\" tabindex=\"-1\">Trigonometric Integrals<\/a>\n\t<ul class=\"sub-menu elementor-nav-menu--dropdown\">\n\t\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-19925\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/integration\/problem-set-integration\/integration-practice-problems-trigonometric-integrals\/\" class=\"elementor-sub-item\" tabindex=\"-1\">Problem Set<\/a><\/li>\n\t\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-19954\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/integration\/problem-set-integration\/virtual-lessons-integration-trigonometric-integrals\/\" class=\"elementor-sub-item\" tabindex=\"-1\">Virtual Lessons<\/a><\/li>\n\t<\/ul>\n<\/li>\n\t<li class=\"menu-item menu-item-type-custom menu-item-object-custom menu-item-has-children menu-item-19886\"><a href=\"#\" class=\"elementor-sub-item elementor-item-anchor\" tabindex=\"-1\">Trigonometric Substitution<\/a>\n\t<ul class=\"sub-menu elementor-nav-menu--dropdown\">\n\t\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-19926\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/integration\/problem-set-integration\/integration-practice-problems-trigonometric-substitution\/\" class=\"elementor-sub-item\" tabindex=\"-1\">Problem Set<\/a><\/li>\n\t\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-19953\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/integration\/problem-set-integration\/virtual-lessons-integration-trigonometric-substitution\/\" class=\"elementor-sub-item\" tabindex=\"-1\">Virtual Lessons<\/a><\/li>\n\t<\/ul>\n<\/li>\n\t<li class=\"menu-item menu-item-type-custom menu-item-object-custom menu-item-has-children menu-item-19884\"><a href=\"#\" class=\"elementor-sub-item elementor-item-anchor\" tabindex=\"-1\">Partial Fractions<\/a>\n\t<ul class=\"sub-menu elementor-nav-menu--dropdown\">\n\t\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-19924\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/integration\/problem-set-integration\/integration-practice-problems-partial-fractions\/\" class=\"elementor-sub-item\" tabindex=\"-1\">Problem Set<\/a><\/li>\n\t\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-19955\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/integration\/problem-set-integration\/virtual-lessons-integration-partialfractions\/\" class=\"elementor-sub-item\" tabindex=\"-1\">Virtual Lessons<\/a><\/li>\n\t<\/ul>\n<\/li>\n\t<li class=\"menu-item menu-item-type-custom menu-item-object-custom menu-item-has-children menu-item-19893\"><a href=\"#\" class=\"elementor-sub-item elementor-item-anchor\" tabindex=\"-1\">Improper Integrals<\/a>\n\t<ul class=\"sub-menu elementor-nav-menu--dropdown\">\n\t\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-19927\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/integration\/problem-set-integration\/integration-practice-problems-improper-integrals\/\" class=\"elementor-sub-item\" tabindex=\"-1\">Problem Set<\/a><\/li>\n\t\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-19966\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/integration\/problem-set-integration\/virtual-lessons-integration-improper-integrals\/\" class=\"elementor-sub-item\" tabindex=\"-1\">Virtual Lessons<\/a><\/li>\n\t<\/ul>\n<\/li>\n<\/ul>\n<\/li>\n<li class=\"menu-item menu-item-type-custom menu-item-object-custom menu-item-has-children menu-item-19578\"><a href=\"#\" class=\"elementor-item elementor-item-anchor\" tabindex=\"-1\">Integration Applications<\/a>\n<ul class=\"sub-menu elementor-nav-menu--dropdown\">\n\t<li class=\"menu-item menu-item-type-custom menu-item-object-custom menu-item-has-children menu-item-19888\"><a href=\"#\" class=\"elementor-sub-item elementor-item-anchor\" tabindex=\"-1\">Area<\/a>\n\t<ul class=\"sub-menu elementor-nav-menu--dropdown\">\n\t\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-19928\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/integration\/problem-set-integration\/integration-practice-problems-application-area\/\" class=\"elementor-sub-item\" tabindex=\"-1\">Problem Set<\/a><\/li>\n\t\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-20136\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/integration\/problem-set-integration\/virtual-lessons-integration-area\/\" class=\"elementor-sub-item\" tabindex=\"-1\">Virtual Lessons<\/a><\/li>\n\t<\/ul>\n<\/li>\n\t<li class=\"menu-item menu-item-type-custom menu-item-object-custom menu-item-has-children menu-item-19889\"><a href=\"#\" class=\"elementor-sub-item elementor-item-anchor\" tabindex=\"-1\">Volumes [Nonrotational]<\/a>\n\t<ul class=\"sub-menu elementor-nav-menu--dropdown\">\n\t\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-19929\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/integration\/problem-set-integration\/integration-practice-problems-application-volume\/\" class=\"elementor-sub-item\" tabindex=\"-1\">Problem Set<\/a><\/li>\n\t\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-20582\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/integration\/problem-set-integration\/virtual-lessons-integration-trigonometric-substitution\/\" class=\"elementor-sub-item\" tabindex=\"-1\">Virtual Lessons<\/a><\/li>\n\t<\/ul>\n<\/li>\n\t<li class=\"menu-item menu-item-type-custom menu-item-object-custom menu-item-has-children menu-item-19890\"><a href=\"#\" class=\"elementor-sub-item elementor-item-anchor\" tabindex=\"-1\">Volume [Rotational]<\/a>\n\t<ul class=\"sub-menu elementor-nav-menu--dropdown\">\n\t\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-19931\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/integration\/problem-set-integration\/integration-practice-problems-application-rotational-volumes\/\" class=\"elementor-sub-item\" tabindex=\"-1\">Problem Set<\/a><\/li>\n\t\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-20160\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/integration\/problem-set-integration\/virtual-lessons-integration-rotational-volume\/\" class=\"elementor-sub-item\" tabindex=\"-1\">Virtual Lessons<\/a><\/li>\n\t<\/ul>\n<\/li>\n\t<li class=\"menu-item menu-item-type-custom menu-item-object-custom menu-item-has-children menu-item-19891\"><a href=\"#\" class=\"elementor-sub-item elementor-item-anchor\" tabindex=\"-1\">Surface Area<\/a>\n\t<ul class=\"sub-menu elementor-nav-menu--dropdown\">\n\t\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-19930\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/integration\/problem-set-integration\/integration-practice-problems-application-surface-area\/\" class=\"elementor-sub-item\" tabindex=\"-1\">Problem Set<\/a><\/li>\n\t\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-20165\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/integration\/problem-set-integration\/virtual-lessons-integration-surface-area\/\" class=\"elementor-sub-item\" tabindex=\"-1\">Virtual Lessons<\/a><\/li>\n\t<\/ul>\n<\/li>\n\t<li class=\"menu-item menu-item-type-custom menu-item-object-custom menu-item-has-children menu-item-19892\"><a href=\"#\" class=\"elementor-sub-item elementor-item-anchor\" tabindex=\"-1\">Arc Length<\/a>\n\t<ul class=\"sub-menu elementor-nav-menu--dropdown\">\n\t\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-20177\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/integration\/problem-set-integration\/integration-practice-problems-arc-length\/\" class=\"elementor-sub-item\" tabindex=\"-1\">Problem Set<\/a><\/li>\n\t\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-20176\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/integration\/problem-set-integration\/virtual-lessons-integration-arc-length\/\" class=\"elementor-sub-item\" tabindex=\"-1\">Virtual Lessons<\/a><\/li>\n\t<\/ul>\n<\/li>\n<\/ul>\n<\/li>\n<li class=\"menu-item menu-item-type-custom menu-item-object-custom menu-item-has-children menu-item-19579\"><a href=\"#\" class=\"elementor-item elementor-item-anchor\" tabindex=\"-1\">Numerical Integration<\/a>\n<ul class=\"sub-menu elementor-nav-menu--dropdown\">\n\t<li class=\"menu-item menu-item-type-custom menu-item-object-custom menu-item-has-children menu-item-19894\"><a href=\"#\" class=\"elementor-sub-item elementor-item-anchor\" tabindex=\"-1\">Reimann Sums<\/a>\n\t<ul class=\"sub-menu elementor-nav-menu--dropdown\">\n\t\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-20110\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/integration\/problem-set-integration\/integration-practice-problems-riemann-sums\/\" class=\"elementor-sub-item\" tabindex=\"-1\">Problem Set<\/a><\/li>\n\t\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-20128\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/integration\/problem-set-integration\/virtual-lessons-integration-riemann-sum\/\" class=\"elementor-sub-item\" tabindex=\"-1\">Virtual Lessons<\/a><\/li>\n\t<\/ul>\n<\/li>\n\t<li class=\"menu-item menu-item-type-custom menu-item-object-custom menu-item-has-children menu-item-19895\"><a href=\"#\" class=\"elementor-sub-item elementor-item-anchor\" tabindex=\"-1\">Trapezoidal Rule<\/a>\n\t<ul class=\"sub-menu elementor-nav-menu--dropdown\">\n\t\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-20109\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/integration\/problem-set-integration\/integration-practice-problems-trapezoidal\/\" class=\"elementor-sub-item\" tabindex=\"-1\">Problem Set<\/a><\/li>\n\t\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-20138\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/integration\/problem-set-integration\/virtual-lessons-integration-trapezoidal\/\" class=\"elementor-sub-item\" tabindex=\"-1\">Virtual Lessons<\/a><\/li>\n\t<\/ul>\n<\/li>\n\t<li class=\"menu-item menu-item-type-custom menu-item-object-custom menu-item-has-children menu-item-19896\"><a href=\"#\" class=\"elementor-sub-item elementor-item-anchor\" tabindex=\"-1\">Simpson&#8217;s Rule<\/a>\n\t<ul class=\"sub-menu elementor-nav-menu--dropdown\">\n\t\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-20108\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/integration\/problem-set-integration\/integration-practice-problems-simpsons\/\" class=\"elementor-sub-item\" tabindex=\"-1\">Problem Set<\/a><\/li>\n\t\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-20137\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/integration\/problem-set-integration\/virtual-lessons-integration-simpsons\/\" class=\"elementor-sub-item\" tabindex=\"-1\">Virtual Lessons<\/a><\/li>\n\t<\/ul>\n<\/li>\n<\/ul>\n<\/li>\n<\/ul>\t\t\t<\/nav>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-a75a15d elementor-section-stretched elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"a75a15d\" data-element_type=\"section\" data-settings=\"{&quot;stretch_section&quot;:&quot;section-stretched&quot;}\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-555d8f5\" data-id=\"555d8f5\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-356b90a elementor-widget-divider--view-line elementor-widget elementor-widget-divider\" data-id=\"356b90a\" data-element_type=\"widget\" data-widget_type=\"divider.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<div class=\"elementor-divider\">\n\t\t\t<span class=\"elementor-divider-separator\">\n\t\t\t\t\t\t<\/span>\n\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-6f95eaf elementor-widget elementor-widget-text-editor\" data-id=\"6f95eaf\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p style=\"text-align: justify;\"><span style=\"font-size: 20px;\">To determine the volume of rotational symmetries there are two main protocols to follow: the Washer\/Disk Method or the Shell Method. The washer method is a generalization of the disk method. Begin by sketching a 2D model, and <strong>DO NOT<\/strong> forget to include the cross-section. Most students are unable to solve these problems because they forgot to include this part. The cross-section is important for various reasons. Not only does it quickly help you determine which method to use, it also helps to measure heights and radii.\u00a0<\/span><\/p><p style=\"text-align: justify;\"><span style=\"font-size: 20px;\">To compute the radius, always measure <strong>FROM<\/strong> (not TO but FROM) the <strong>axis of the rotation<\/strong>.\u00a0<\/span><\/p><ul><li><span style=\"font-size: 20px;\">Washer\/Disk Method<\/span><ul><li><span style=\"font-size: 20px;\">Cross-Sections are taken perpendicular to the Axis of Rotation<\/span><\/li><\/ul><\/li><li><span style=\"font-size: 20px;\">(Cylindrical) Shell Method<\/span><ul><li><span style=\"font-size: 20px;\">Cross-Sections are taken parallel to the Axis of Rotation<\/span><\/li><\/ul><\/li><\/ul>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-40880ff elementor-widget-divider--view-line elementor-widget elementor-widget-divider\" data-id=\"40880ff\" data-element_type=\"widget\" data-widget_type=\"divider.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<div class=\"elementor-divider\">\n\t\t\t<span class=\"elementor-divider-separator\">\n\t\t\t\t\t\t<\/span>\n\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-a99c65d elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"a99c65d\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-e0f1a78\" data-id=\"e0f1a78\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-fb6564e\" data-id=\"fb6564e\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-48678aa elementor-view-framed elementor-position-left elementor-shape-circle elementor-mobile-position-top elementor-vertical-align-top elementor-widget elementor-widget-icon-box\" data-id=\"48678aa\" data-element_type=\"widget\" data-widget_type=\"icon-box.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<div class=\"elementor-icon-box-wrapper\">\n\n\t\t\t\t\t\t<div class=\"elementor-icon-box-icon\">\n\t\t\t\t<a href=\"https:\/\/onlinemathtutor.co\/index.php\/course-content\/calculus-resources\/integration\/problem-set-integration\/virtual-lessons-integration-rotational-volume\/?playlist=b24a5c0&#038;video=1adcc07\" class=\"elementor-icon elementor-animation-grow\" tabindex=\"-1\" aria-label=\"Virtual Lessons\">\n\t\t\t\t<i aria-hidden=\"true\" class=\"fas fa-chalkboard-teacher\"><\/i>\t\t\t\t<\/a>\n\t\t\t<\/div>\n\t\t\t\n\t\t\t\t\t\t<div class=\"elementor-icon-box-content\">\n\n\t\t\t\t\t\t\t\t\t<h2 class=\"elementor-icon-box-title\">\n\t\t\t\t\t\t<a href=\"https:\/\/onlinemathtutor.co\/index.php\/course-content\/calculus-resources\/integration\/problem-set-integration\/virtual-lessons-integration-rotational-volume\/?playlist=b24a5c0&#038;video=1adcc07\" >\n\t\t\t\t\t\t\tVirtual Lessons\t\t\t\t\t\t<\/a>\n\t\t\t\t\t<\/h2>\n\t\t\t\t\n\t\t\t\t\t\t\t\t\t<p class=\"elementor-icon-box-description\">\n\t\t\t\t\t\tNeed some additional help understanding how to apply this integration technique? Click Here to visit the virtual lesson section. \t\t\t\t\t<\/p>\n\t\t\t\t\n\t\t\t<\/div>\n\t\t\t\n\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-9bfaa1b\" data-id=\"9bfaa1b\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-b1c245c elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"b1c245c\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-db446e9\" data-id=\"db446e9\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-86d2d68 elementor-widget-divider--view-line elementor-widget elementor-widget-divider\" data-id=\"86d2d68\" data-element_type=\"widget\" data-widget_type=\"divider.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<div class=\"elementor-divider\">\n\t\t\t<span class=\"elementor-divider-separator\">\n\t\t\t\t\t\t<\/span>\n\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-b9b6a13 elementor-widget elementor-widget-spacer\" data-id=\"b9b6a13\" data-element_type=\"widget\" data-widget_type=\"spacer.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<div class=\"elementor-spacer\">\n\t\t\t<div class=\"elementor-spacer-inner\"><\/div>\n\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-7c20c1d elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"7c20c1d\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-6c836f6\" data-id=\"6c836f6\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-f33923c elementor-widget elementor-widget-heading\" data-id=\"f33923c\" data-element_type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<h2 class=\"elementor-heading-title elementor-size-large\">Practice Problems<\/h2>\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-8562456 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"8562456\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-a9bedd6\" data-id=\"a9bedd6\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-0026ba1 elementor-widget elementor-widget-text-editor\" data-id=\"0026ba1\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p style=\"text-align: justify;\"><span style=\"font-size: 18px;\">Determine the volume for the solid by integrating congruent cross-sections.<\/span><\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-fbb814b elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"fbb814b\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-4c080e4\" data-id=\"4c080e4\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-1a15745 elementor-widget elementor-widget-text-editor\" data-id=\"1a15745\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<ol start=\"1\"><li style=\"margin-bottom: 30px;\">\\(\\displaystyle \\begin{align}y &amp;= x^{-1} \\\\ y&amp; = 0\\\\ 1\\leq &amp;x\\leq 4\\end{align}\\)<\/li><li style=\"margin-bottom: 30px;\">\\(\\displaystyle \\begin{align}y&amp;= \\texttt{ln}(x) \\\\y &amp;= 1\\\\ 0\\leq &amp;x\\leq \\texttt{e}\\end{align} \\)<\/li><li style=\"margin-bottom: 30px;\">\\(\\displaystyle \\begin{align}y &amp;= 2x \\\\y &amp;= x+1 \\\\ 0\\leq &amp;x\\leq 1 \\end{align} \\)<\/li><li style=\"margin-bottom: 30px;\">\\(\\displaystyle \\begin{align} f(x) &amp;= x^{2\/3} \\\\ g(x) &amp;= x^{3\/2} \\\\\u00a0 0\\leq &amp;x \\leq 1 \\end{align} \\)<\/li><\/ol>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-f52c0f4\" data-id=\"f52c0f4\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-0cc621c elementor-widget elementor-widget-text-editor\" data-id=\"0cc621c\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<ol start=\"5\"><li style=\"margin-bottom: 30px;\">\\(\\displaystyle \\begin{align}f(x) &amp;= \\texttt{arcsin}(x) \\\\ 0\\leq &amp;y\\leq 1 \\end{align}\\)<\/li><li style=\"margin-bottom: 30px;\">\\(\\displaystyle \\begin{align} r^2 &amp;= (x &#8211; a)^2 + y^2\u00a0 \\\\ V &amp;= 2{\\pi}^2 ar^2 \\end{align}\\)<\/li><li style=\"margin-bottom: 30px;\">\\(\\displaystyle \\begin{align}f(x) &amp;= 4 &#8211; x^2 \\\\ g(x) &amp;= 2- x \\\\\u00a0 0\\leq &amp;x\\leq 2 \\end{align}\\)<\/li><li style=\"margin-bottom: 30px;\">\\(\\displaystyle \\begin{align} f(x) &amp;= x^3 \\\\ g(x) &amp;= x \\\\ 0\\leq &amp;x \\leq 1 \\end{align}\\)<\/li><\/ol>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-b9d4f88\" data-id=\"b9d4f88\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-1ee2235 elementor-widget elementor-widget-text-editor\" data-id=\"1ee2235\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<ol start=\"9\"><li style=\"margin-bottom: 30px;\">\\(\\displaystyle \\begin{align}f(x) &amp;= (x-2)^3 \\\\ g(x) &amp;= x- 2 \\\\ 1\\leq x\\leq 2 \\end{align} \\)<\/li><li style=\"margin-bottom: 30px;\">\\(\\displaystyle 2\\pi \\int^1_0 (2x-x^2) dx \\)<\/li><li style=\"margin-bottom: 30px;\">\\(\\displaystyle 2\\pi \\int^1_0 (y+2)\\left(\\sqrt{1-y^2}\\right) dy \\)<\/li><\/ol>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-96a1c60 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"96a1c60\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-83bd0b7\" data-id=\"83bd0b7\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-e0fe46a elementor-align-left elementor-icon-list--layout-traditional elementor-list-item-link-full_width elementor-widget elementor-widget-icon-list\" data-id=\"e0fe46a\" data-element_type=\"widget\" data-widget_type=\"icon-list.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<ul class=\"elementor-icon-list-items\">\n\t\t\t\t\t\t\t<li class=\"elementor-icon-list-item\">\n\t\t\t\t\t\t\t\t\t\t\t<a href=\"https:\/\/onlinemathtutor.co\/index.php\/problem-set-integration\/\">\n\n\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-icon-list-icon\">\n\t\t\t\t\t\t\t<i aria-hidden=\"true\" class=\"fas fa-arrow-circle-right\"><\/i>\t\t\t\t\t\t<\/span>\n\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-icon-list-text\">Back To Main Page: Integration Problem Sets<\/span>\n\t\t\t\t\t\t\t\t\t\t\t<\/a>\n\t\t\t\t\t\t\t\t\t<\/li>\n\t\t\t\t\t\t<\/ul>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-a9e2d37 elementor-widget elementor-widget-spacer\" data-id=\"a9e2d37\" data-element_type=\"widget\" data-widget_type=\"spacer.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<div class=\"elementor-spacer\">\n\t\t\t<div class=\"elementor-spacer-inner\"><\/div>\n\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-9d0490a elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"9d0490a\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-09b8a9e\" data-id=\"09b8a9e\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-9422751 elementor-widget elementor-widget-heading\" data-id=\"9422751\" data-element_type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<h2 class=\"elementor-heading-title elementor-size-large\">Suggestive Solution Guide<\/h2>\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-58774df1 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"58774df1\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-5522fb6e\" data-id=\"5522fb6e\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-793c7ec elementor-widget elementor-widget-toggle\" data-id=\"793c7ec\" data-element_type=\"widget\" data-widget_type=\"toggle.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<div class=\"elementor-toggle\">\n\t\t\t\t\t\t\t<div class=\"elementor-toggle-item\">\n\t\t\t\t\t<h2 id=\"elementor-tab-title-1271\" class=\"elementor-tab-title\" data-tab=\"1\" role=\"button\" aria-controls=\"elementor-tab-content-1271\" aria-expanded=\"false\">\n\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon elementor-toggle-icon-left\" aria-hidden=\"true\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon-closed\"><i class=\"fas fa-toggle-off\"><\/i><\/span>\n\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon-opened\"><i class=\"elementor-toggle-icon-opened fas fa-toggle-on\"><\/i><\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t\t<\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t<a class=\"elementor-toggle-title\" tabindex=\"0\">Problem 1: Solution<\/a>\n\t\t\t\t\t<\/h2>\n\n\t\t\t\t\t<div id=\"elementor-tab-content-1271\" class=\"elementor-tab-content elementor-clearfix\" data-tab=\"1\" role=\"region\" aria-labelledby=\"elementor-tab-title-1271\"><h3>Sketch the indicated region bounded by the given graphs and compute the volume that is generated by rotating the region about the x-axis:<\/h3><h3>\u00a0<\/h3><h3><img fetchpriority=\"high\" decoding=\"async\" class=\"wp-image-6354 size-medium aligncenter\" src=\"https:\/\/bisonmath.com\/wp-content\/uploads\/2022\/03\/Screenshot-19-300x300.png\" alt=\"\" width=\"300\" height=\"300\" \/><\/h3><h4>Sketeched area to be rotated around the x-axis. Given how we already have the endoints and area under the curve, the only thing left is to select the appropiate method of integration. Given how we will rotate the area around the x-axis and the differential segment is perpendicular to the axis of rotation without any airgap, we choose the disk mehtod. Let the radius defined by the length of the segment be the function itself \\(\\left( y = \\frac{1}{x} \\right)\\)<\/h4>\\(<br \/>\\begin{align*}<br \/>&amp; V = \\pi \\int^4_1 \\left( \\frac{1}{x} \\right)^2 dx = \\pi \\int^4_1 \\frac{1}{x^2} dx = \\pi \\left[ \\frac{-1}{x} \\right] \\Bigg\\rvert^4_1 = \\pi \\left[ \\left(\\frac{-1}{4}\\right) &#8211; \\left(\\frac{-1}{1}\\right)\\right] = \\pi \\left[ \\left(\\frac{-1}{4}\\right) + 1 \\right]\u00a0\\\\[5pt]<br \/>&amp; \\color{green}{V = \\frac{3}{4}\\pi}<br \/>\\end{align*}<br \/>\\)<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t\t<script type=\"application\/ld+json\">{\"@context\":\"https:\\\/\\\/schema.org\",\"@type\":\"FAQPage\",\"mainEntity\":[{\"@type\":\"Question\",\"name\":\"Problem 1: Solution\",\"acceptedAnswer\":{\"@type\":\"Answer\",\"text\":\"<h3>Sketch the indicated region bounded by the given graphs and compute the volume that is generated by rotating the region about the x-axis:<\\\/h3><h3>\\u00a0<\\\/h3><h3><img class=\\\"wp-image-6354 size-medium aligncenter\\\" src=\\\"https:\\\/\\\/bisonmath.com\\\/wp-content\\\/uploads\\\/2022\\\/03\\\/Screenshot-19-300x300.png\\\" alt=\\\"\\\" width=\\\"300\\\" height=\\\"300\\\" \\\/><\\\/h3><h4>Sketeched area to be rotated around the x-axis. Given how we already have the endoints and area under the curve, the only thing left is to select the appropiate method of integration. Given how we will rotate the area around the x-axis and the differential segment is perpendicular to the axis of rotation without any airgap, we choose the disk mehtod. Let the radius defined by the length of the segment be the function itself \\\\(\\\\left( y = \\\\frac{1}{x} \\\\right)\\\\)<\\\/h4>\\\\(<br \\\/>\\\\begin{align*}<br \\\/>&amp; V = \\\\pi \\\\int^4_1 \\\\left( \\\\frac{1}{x} \\\\right)^2 dx = \\\\pi \\\\int^4_1 \\\\frac{1}{x^2} dx = \\\\pi \\\\left[ \\\\frac{-1}{x} \\\\right] \\\\Bigg\\\\rvert^4_1 = \\\\pi \\\\left[ \\\\left(\\\\frac{-1}{4}\\\\right) &#8211; \\\\left(\\\\frac{-1}{1}\\\\right)\\\\right] = \\\\pi \\\\left[ \\\\left(\\\\frac{-1}{4}\\\\right) + 1 \\\\right]\\u00a0\\\\\\\\[5pt]<br \\\/>&amp; \\\\color{green}{V = \\\\frac{3}{4}\\\\pi}<br \\\/>\\\\end{align*}<br \\\/>\\\\)\"}}]}<\/script>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-436fb7c5 elementor-widget elementor-widget-toggle\" data-id=\"436fb7c5\" data-element_type=\"widget\" data-widget_type=\"toggle.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<div class=\"elementor-toggle\">\n\t\t\t\t\t\t\t<div class=\"elementor-toggle-item\">\n\t\t\t\t\t<h2 id=\"elementor-tab-title-1131\" class=\"elementor-tab-title\" data-tab=\"1\" role=\"button\" aria-controls=\"elementor-tab-content-1131\" aria-expanded=\"false\">\n\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon elementor-toggle-icon-left\" aria-hidden=\"true\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon-closed\"><i class=\"fas fa-toggle-off\"><\/i><\/span>\n\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon-opened\"><i class=\"elementor-toggle-icon-opened fas fa-toggle-on\"><\/i><\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t\t<\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t<a class=\"elementor-toggle-title\" tabindex=\"0\">Problem 2: Solution<\/a>\n\t\t\t\t\t<\/h2>\n\n\t\t\t\t\t<div id=\"elementor-tab-content-1131\" class=\"elementor-tab-content elementor-clearfix\" data-tab=\"1\" role=\"region\" aria-labelledby=\"elementor-tab-title-1131\"><h3>Sketch the indicated region bounded by the given graphs and compute the volume that is generated by rotating the region about the x-axis:<\/h3><p><img decoding=\"async\" class=\"size-medium wp-image-6384 aligncenter\" src=\"https:\/\/bisonmath.com\/wp-content\/uploads\/2022\/03\/Screenshot-22-300x300.png\" alt=\"\" width=\"300\" height=\"300\" \/><\/p><h4>Sketched area to be rotated around the x-axis. Because we already know that the area is going to be rotated around the x-axis, then the graphic can&#8217;t go beyond said rotational axis. This is why we have to split the integration area into two different integrals (where the changes are the endpoints and the radius. First, let&#8217;s calulate the volume of the pink-shaded area using the disk method:<\/h4>\\(<br \/>\\begin{align*}<br \/>&amp; V = \\pi \\int^1_0 (1)^2 dx = \\pi[x]\\bigg\\rvert^1_0 = \\pi[ 1 -0] = \\pi<br \/>\\end{align*}<br \/>\\)<h4>Now, for the second area (shaded in blue), use the washer method because of the airgap between the axis of rotation and the area to be rotated. Let \\(R = 1, r = \\texttt{ln}(x)\\), and the integration limits be \\(1 \\le x \\le \\texttt{e}\\). Using this data yields:<\/h4>\\(<br \/>\\begin{align*}<br \/>&amp; V = \\pi \\int^{\\texttt{e}}_1 \\left[ (1)^2 &#8211; \\left(\\texttt{ln}(x)\\right)^2 \\right] dx<br \/>\\end{align*}<br \/>\\)<h4>In order to solve for the second integrand, use IBP. Let \\(u=\\left(\\texttt{ln}(x)\\right)^2\\) and \\(dv = dx\\), so \\(du = \\frac{2\\texttt{ln}(x)}{x}dx\\) and \\(v = x\\). Applying those substitutions yields:<\/h4>\\(<br \/>\\begin{align*}<br \/>\\int \\left(\\texttt{ln}(x)\\right)^2 dx = x\\texttt{ln}^2(x) \\:- 2\\int \\frac{\\texttt{ln}(x)}{x}(x) dx = x\\texttt{ln}^2(x) \\:- 2\\int \\texttt{Ln}(x)dx<br \/>\\end{align*}<br \/>\\)<h4>Apply IBP once again. Let \\(u=\\texttt{ln}(x)\\) and \\(dv = dx\\), so \\(du = \\frac{1}{x}dx\\) and \\(v = x\\). Applying those substitutions yields:<\/h4>\\(<br \/>\\begin{align*}<br \/>x\\texttt{ln}^2(x) \\:- 2\\int \\texttt{ln}(x)dx<br \/>&amp; = x\\texttt{ln}^2(x) \\:- 2\\left[x\\texttt{ln}(x) \\:- \\int \\frac{1}{x}(x)dx\\right] \\\\[5pt]<br \/>&amp; = x\\texttt{ln}^2(x) \\:- 2x\\texttt{ln}(x) \\:+ 2\\int dx \\\\[5pt]<br \/>&amp; = x\\texttt{ln}^2(x) \\:- 2x\\texttt{ln}(x) \\:+ 2x \\\\[5pt]<br \/>\\end{align*}<br \/>\\)<h4>Use the previous result to find the final volume:<\/h4>\\(<br \/>\\begin{align*}<br \/>&amp; V = \\pi \\int^{\\texttt{e}}_1 \\left[ (1)^2 &#8211; \\left(\\texttt{ln}(x)\\right)^2 \\right] dx \\\\[5pt]<br \/>&amp; V = \\pi \\left[ x \\:- \\left( x\\texttt{ln}^2(x) \\:- 2x\\texttt{Ln}(x) \\:+ 2x \\right) \\right] \\Bigg\\rvert^{\\texttt{e}}_1<br \/>= \\pi \\left[ x \\:- x\\texttt{ln}^2(x) \\:+ 2x\\texttt{ln}(x) \\:- 2x \\right] \\Bigg\\rvert^{\\texttt{e}}_1 \\\\[5pt]<br \/>&amp; V = \\pi \\left[ \\left(\\texttt{e} \\:- \\texttt{e}\\texttt{ln}^2(\\texttt{e}) \\:+ 2\\texttt{e}\\texttt{ln}(\\texttt{e}) \\:- 2\\texttt{e}\\right) \\:- \\left(1 \\:- \\texttt{ln}^2(1) \\:+ 2\\texttt{ln}(1) \\:- 2\\right) \\right] \\\\[5pt]<br \/>&amp; V = \\pi [(\\texttt{e} \\:- \\texttt{e} \\:+ 2\\texttt{e} \\:- 2\\texttt{e}) \\:- (1 \\:- 0 + 0 &#8211; 2)] \\\\[5pt]<br \/>&amp; V = \\pi[2 \\:- 1] = \\pi<br \/>\\end{align*}<br \/>\\)<h4>Finally, combine the results of both definite integrals to obtain the final answer:<\/h4>\\(<br \/>\\begin{align*}<br \/>&amp; V = \\pi \\int^1_0 (1)^2 dx \\:+ \\pi \\int^{\\texttt{e}}_1 \\left[ (1)^2 &#8211; \\left(\\texttt{ln}(x)\\right)^2 \\right] dx = \\pi + \\pi\\\\[5pt]<br \/>&amp; \\color{green}{V = 2\\pi}<br \/>\\end{align*}<br \/>\\)<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t\t<script type=\"application\/ld+json\">{\"@context\":\"https:\\\/\\\/schema.org\",\"@type\":\"FAQPage\",\"mainEntity\":[{\"@type\":\"Question\",\"name\":\"Problem 2: Solution\",\"acceptedAnswer\":{\"@type\":\"Answer\",\"text\":\"<h3>Sketch the indicated region bounded by the given graphs and compute the volume that is generated by rotating the region about the x-axis:<\\\/h3><p><img class=\\\"size-medium wp-image-6384 aligncenter\\\" src=\\\"https:\\\/\\\/bisonmath.com\\\/wp-content\\\/uploads\\\/2022\\\/03\\\/Screenshot-22-300x300.png\\\" alt=\\\"\\\" width=\\\"300\\\" height=\\\"300\\\" \\\/><\\\/p><h4>Sketched area to be rotated around the x-axis. Because we already know that the area is going to be rotated around the x-axis, then the graphic can&#8217;t go beyond said rotational axis. This is why we have to split the integration area into two different integrals (where the changes are the endpoints and the radius. First, let&#8217;s calulate the volume of the pink-shaded area using the disk method:<\\\/h4>\\\\(<br \\\/>\\\\begin{align*}<br \\\/>&amp; V = \\\\pi \\\\int^1_0 (1)^2 dx = \\\\pi[x]\\\\bigg\\\\rvert^1_0 = \\\\pi[ 1 -0] = \\\\pi<br \\\/>\\\\end{align*}<br \\\/>\\\\)<h4>Now, for the second area (shaded in blue), use the washer method because of the airgap between the axis of rotation and the area to be rotated. Let \\\\(R = 1, r = \\\\texttt{ln}(x)\\\\), and the integration limits be \\\\(1 \\\\le x \\\\le \\\\texttt{e}\\\\). Using this data yields:<\\\/h4>\\\\(<br \\\/>\\\\begin{align*}<br \\\/>&amp; V = \\\\pi \\\\int^{\\\\texttt{e}}_1 \\\\left[ (1)^2 &#8211; \\\\left(\\\\texttt{ln}(x)\\\\right)^2 \\\\right] dx<br \\\/>\\\\end{align*}<br \\\/>\\\\)<h4>In order to solve for the second integrand, use IBP. Let \\\\(u=\\\\left(\\\\texttt{ln}(x)\\\\right)^2\\\\) and \\\\(dv = dx\\\\), so \\\\(du = \\\\frac{2\\\\texttt{ln}(x)}{x}dx\\\\) and \\\\(v = x\\\\). Applying those substitutions yields:<\\\/h4>\\\\(<br \\\/>\\\\begin{align*}<br \\\/>\\\\int \\\\left(\\\\texttt{ln}(x)\\\\right)^2 dx = x\\\\texttt{ln}^2(x) \\\\:- 2\\\\int \\\\frac{\\\\texttt{ln}(x)}{x}(x) dx = x\\\\texttt{ln}^2(x) \\\\:- 2\\\\int \\\\texttt{Ln}(x)dx<br \\\/>\\\\end{align*}<br \\\/>\\\\)<h4>Apply IBP once again. Let \\\\(u=\\\\texttt{ln}(x)\\\\) and \\\\(dv = dx\\\\), so \\\\(du = \\\\frac{1}{x}dx\\\\) and \\\\(v = x\\\\). Applying those substitutions yields:<\\\/h4>\\\\(<br \\\/>\\\\begin{align*}<br \\\/>x\\\\texttt{ln}^2(x) \\\\:- 2\\\\int \\\\texttt{ln}(x)dx<br \\\/>&amp; = x\\\\texttt{ln}^2(x) \\\\:- 2\\\\left[x\\\\texttt{ln}(x) \\\\:- \\\\int \\\\frac{1}{x}(x)dx\\\\right] \\\\\\\\[5pt]<br \\\/>&amp; = x\\\\texttt{ln}^2(x) \\\\:- 2x\\\\texttt{ln}(x) \\\\:+ 2\\\\int dx \\\\\\\\[5pt]<br \\\/>&amp; = x\\\\texttt{ln}^2(x) \\\\:- 2x\\\\texttt{ln}(x) \\\\:+ 2x \\\\\\\\[5pt]<br \\\/>\\\\end{align*}<br \\\/>\\\\)<h4>Use the previous result to find the final volume:<\\\/h4>\\\\(<br \\\/>\\\\begin{align*}<br \\\/>&amp; V = \\\\pi \\\\int^{\\\\texttt{e}}_1 \\\\left[ (1)^2 &#8211; \\\\left(\\\\texttt{ln}(x)\\\\right)^2 \\\\right] dx \\\\\\\\[5pt]<br \\\/>&amp; V = \\\\pi \\\\left[ x \\\\:- \\\\left( x\\\\texttt{ln}^2(x) \\\\:- 2x\\\\texttt{Ln}(x) \\\\:+ 2x \\\\right) \\\\right] \\\\Bigg\\\\rvert^{\\\\texttt{e}}_1<br \\\/>= \\\\pi \\\\left[ x \\\\:- x\\\\texttt{ln}^2(x) \\\\:+ 2x\\\\texttt{ln}(x) \\\\:- 2x \\\\right] \\\\Bigg\\\\rvert^{\\\\texttt{e}}_1 \\\\\\\\[5pt]<br \\\/>&amp; V = \\\\pi \\\\left[ \\\\left(\\\\texttt{e} \\\\:- \\\\texttt{e}\\\\texttt{ln}^2(\\\\texttt{e}) \\\\:+ 2\\\\texttt{e}\\\\texttt{ln}(\\\\texttt{e}) \\\\:- 2\\\\texttt{e}\\\\right) \\\\:- \\\\left(1 \\\\:- \\\\texttt{ln}^2(1) \\\\:+ 2\\\\texttt{ln}(1) \\\\:- 2\\\\right) \\\\right] \\\\\\\\[5pt]<br \\\/>&amp; V = \\\\pi [(\\\\texttt{e} \\\\:- \\\\texttt{e} \\\\:+ 2\\\\texttt{e} \\\\:- 2\\\\texttt{e}) \\\\:- (1 \\\\:- 0 + 0 &#8211; 2)] \\\\\\\\[5pt]<br \\\/>&amp; V = \\\\pi[2 \\\\:- 1] = \\\\pi<br \\\/>\\\\end{align*}<br \\\/>\\\\)<h4>Finally, combine the results of both definite integrals to obtain the final answer:<\\\/h4>\\\\(<br \\\/>\\\\begin{align*}<br \\\/>&amp; V = \\\\pi \\\\int^1_0 (1)^2 dx \\\\:+ \\\\pi \\\\int^{\\\\texttt{e}}_1 \\\\left[ (1)^2 &#8211; \\\\left(\\\\texttt{ln}(x)\\\\right)^2 \\\\right] dx = \\\\pi + \\\\pi\\\\\\\\[5pt]<br \\\/>&amp; \\\\color{green}{V = 2\\\\pi}<br \\\/>\\\\end{align*}<br \\\/>\\\\)\"}}]}<\/script>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-5612a5ae elementor-widget elementor-widget-toggle\" data-id=\"5612a5ae\" data-element_type=\"widget\" data-widget_type=\"toggle.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<div class=\"elementor-toggle\">\n\t\t\t\t\t\t\t<div class=\"elementor-toggle-item\">\n\t\t\t\t\t<h2 id=\"elementor-tab-title-1441\" class=\"elementor-tab-title\" data-tab=\"1\" role=\"button\" aria-controls=\"elementor-tab-content-1441\" aria-expanded=\"false\">\n\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon elementor-toggle-icon-left\" aria-hidden=\"true\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon-closed\"><i class=\"fas fa-toggle-off\"><\/i><\/span>\n\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon-opened\"><i class=\"elementor-toggle-icon-opened fas fa-toggle-on\"><\/i><\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t\t<\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t<a class=\"elementor-toggle-title\" tabindex=\"0\">Problem 3: Solution<\/a>\n\t\t\t\t\t<\/h2>\n\n\t\t\t\t\t<div id=\"elementor-tab-content-1441\" class=\"elementor-tab-content elementor-clearfix\" data-tab=\"1\" role=\"region\" aria-labelledby=\"elementor-tab-title-1441\"><h3>Compute the volume generated by rotating the indicated bounded region about the x-axis:<\/h3><p><img decoding=\"async\" class=\"size-medium wp-image-6459 aligncenter\" src=\"https:\/\/bisonmath.com\/wp-content\/uploads\/2022\/03\/Screenshot-24-300x300.png\" alt=\"\" width=\"300\" height=\"300\" \/><\/p><h4>Sketched area to be rotated around the x-axis. Given the existence of a gap between the rotational area and the axis of rotation, we use the washer method. Let R = x + 1 and r = 2x. Applying this data into the integrand yields:<\/h4>\\(<br \/>\\begin{align*}<br \/>&amp; V = \\pi \\int^1_0 \\left[(x+1)^2\\ &#8211; (2x)^2 \\right] dx = \\pi \\int^1_0 (-3x^2 + 2x + 1) dx = \\pi \\left[-x^3 \\:+ x^2 \\:+ x\\right] \\Bigg\\rvert^1_0 \\\\[5pt]<br \/>&amp; V = \\pi\\left[ \\left(-(1)^3 \\:+ 1^2 \\:+ 1\\right) \\:- \\left(-\\frac{0^3}{3} \\:+ 0^2 \\:+ 0\\right) \\right] \\\\[5pt]<br \/>&amp; \\color{green}{V = \\pi}<br \/>\\end{align*}<br \/>\\)<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t\t<script type=\"application\/ld+json\">{\"@context\":\"https:\\\/\\\/schema.org\",\"@type\":\"FAQPage\",\"mainEntity\":[{\"@type\":\"Question\",\"name\":\"Problem 3: Solution\",\"acceptedAnswer\":{\"@type\":\"Answer\",\"text\":\"<h3>Compute the volume generated by rotating the indicated bounded region about the x-axis:<\\\/h3><p><img class=\\\"size-medium wp-image-6459 aligncenter\\\" src=\\\"https:\\\/\\\/bisonmath.com\\\/wp-content\\\/uploads\\\/2022\\\/03\\\/Screenshot-24-300x300.png\\\" alt=\\\"\\\" width=\\\"300\\\" height=\\\"300\\\" \\\/><\\\/p><h4>Sketched area to be rotated around the x-axis. Given the existence of a gap between the rotational area and the axis of rotation, we use the washer method. Let R = x + 1 and r = 2x. Applying this data into the integrand yields:<\\\/h4>\\\\(<br \\\/>\\\\begin{align*}<br \\\/>&amp; V = \\\\pi \\\\int^1_0 \\\\left[(x+1)^2\\\\ &#8211; (2x)^2 \\\\right] dx = \\\\pi \\\\int^1_0 (-3x^2 + 2x + 1) dx = \\\\pi \\\\left[-x^3 \\\\:+ x^2 \\\\:+ x\\\\right] \\\\Bigg\\\\rvert^1_0 \\\\\\\\[5pt]<br \\\/>&amp; V = \\\\pi\\\\left[ \\\\left(-(1)^3 \\\\:+ 1^2 \\\\:+ 1\\\\right) \\\\:- \\\\left(-\\\\frac{0^3}{3} \\\\:+ 0^2 \\\\:+ 0\\\\right) \\\\right] \\\\\\\\[5pt]<br \\\/>&amp; \\\\color{green}{V = \\\\pi}<br \\\/>\\\\end{align*}<br \\\/>\\\\)\"}}]}<\/script>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-4bcf9f6b elementor-widget elementor-widget-toggle\" data-id=\"4bcf9f6b\" data-element_type=\"widget\" data-widget_type=\"toggle.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<div class=\"elementor-toggle\">\n\t\t\t\t\t\t\t<div class=\"elementor-toggle-item\">\n\t\t\t\t\t<h2 id=\"elementor-tab-title-1271\" class=\"elementor-tab-title\" data-tab=\"1\" role=\"button\" aria-controls=\"elementor-tab-content-1271\" aria-expanded=\"false\">\n\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon elementor-toggle-icon-left\" aria-hidden=\"true\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon-closed\"><i class=\"fas fa-toggle-off\"><\/i><\/span>\n\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon-opened\"><i class=\"elementor-toggle-icon-opened fas fa-toggle-on\"><\/i><\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t\t<\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t<a class=\"elementor-toggle-title\" tabindex=\"0\">Problem 4: Solution<\/a>\n\t\t\t\t\t<\/h2>\n\n\t\t\t\t\t<div id=\"elementor-tab-content-1271\" class=\"elementor-tab-content elementor-clearfix\" data-tab=\"1\" role=\"region\" aria-labelledby=\"elementor-tab-title-1271\"><h3>Compute the colume generated by rotating the region bounded by the graphs of the given equations:<\/h3><ul><li><h4>(a) About the x-axis<\/h4><\/li><\/ul><p><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-6492 aligncenter\" src=\"https:\/\/bisonmath.com\/wp-content\/uploads\/2022\/03\/Screenshot-25-300x300.png\" alt=\"\" width=\"300\" height=\"300\" \/><\/p><h4>Sketched area to rotate around the axis. Given the airgap between the shaded area and the axis of rotation, and that the chosen differential is perpendicular to the axis, use the washer method. Let \\(R = x^{\\frac{2}{3}}\\) and \\(r = x^{\\frac{3}{2}}\\). Using this data in the integral yields:<\/h4>\\(<br \/>\\begin{align*}<br \/>&amp; V = \\pi \\int^1_0 \\left[ \\left(x^{\\frac{2}{3}}\\right)^2 \\:- \\left(x^{\\frac{3}{2}}\\right)^2 \\right] dx = \\left[ \\frac{3}{7}x^{\\frac{7}{3}} \\:- \\frac{1}{4}x^4 \\right] \\Bigg\\rvert^1_0 = \\left[ \\frac{3}{7}(1)^{\\frac{7}{3}} \\:- \\frac{1}{4}(1)^4 \\right] \\:- \\left[ \\frac{3}{7}(0)^{\\frac{7}{3}} \\:- \\frac{1}{4}(0)^4 \\right] \\\\[5pt]<br \/>&amp; \\color{green}{V = \\frac{5}{28}\\pi}<br \/>\\end{align*}<br \/>\\)<p>\u00a0<\/p><ul><li><h4>(b) About x=-2<\/h4><\/li><\/ul><p><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-6512 aligncenter\" src=\"https:\/\/bisonmath.com\/wp-content\/uploads\/2022\/03\/Screenshot-26-300x300.png\" alt=\"\" width=\"300\" height=\"300\" \/><\/p><h4>Sketched area and updated axis of rotation. Given how we can&#8217;t change the orientation of the differential (mainly because both ends have to touch both functions), we choose to use the shell method. For that, we need to identify the radius and the height. Let \\(R = x^{\\frac{2}{3}} &#8211; x^{\\frac{3}{2}}\\) and \\(H = x + 2\\). Using this values in the integral yields:<\/h4>\\(<br \/>\\begin{align*}<br \/>&amp; V = 2\\pi \\int^1_0 (x+2) \\left(x^{\\frac{2}{3}} &#8211; x^{\\frac{3}{2}}\\right) dx = 2\\pi \\int^1_0 \\left( x^{\\frac{5}{3}} \\:- x^{\\frac{5}{2}} \\:+ 2x^{\\frac{2}{3}} \\:- 2x^{\\frac{3}{2}} \\right)dx = 2\\pi\\left[ \\frac{3}{8}x^{\\frac{8}{3}} \\:- \\frac{2}{7}x^{\\frac{7}{2}} \\:+ (2)\\frac{3}{5}x^{\\frac{5}{3}} \\:- (2)\\frac{2}{5}x^{\\frac{5}{2}} \\right] \\Bigg\\rvert^1_0 \\\\[5pt]<br \/>&amp; V = 2\\pi \\left( \\left[ \\frac{3}{8}(1)^{\\frac{8}{3}} \\:- \\frac{2}{7}(1)^{\\frac{7}{2}} \\:+ \\frac{6}{5}(1)^{\\frac{5}{3}} \\:- \\frac{4}{5}(1)^{\\frac{5}{2}} \\right] \\:- \\left[ \\frac{3}{8}(0)^{\\frac{8}{3}} \\:- \\frac{2}{7}(0)^{\\frac{7}{2}} \\:+ \\frac{6}{5}(0)^{\\frac{5}{3}} \\:- \\frac{4}{5}(0)^{\\frac{5}{2}} \\right] \\right) \\\\[5pt]<br \/>&amp; V = 2\\pi \\left[ \\frac{3}{8} \\:- \\frac{2}{7} \\:+ \\frac{6}{5} \\:- \\frac{4}{5} \\right] = 2\\pi \\left[\\frac{137}{280}\\right] \\\\[5pt]<br \/>&amp; \\color{green}{V = \\frac{137}{140}\\pi}<br \/>\\end{align*}<br \/>\\)<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t\t<script type=\"application\/ld+json\">{\"@context\":\"https:\\\/\\\/schema.org\",\"@type\":\"FAQPage\",\"mainEntity\":[{\"@type\":\"Question\",\"name\":\"Problem 4: Solution\",\"acceptedAnswer\":{\"@type\":\"Answer\",\"text\":\"<h3>Compute the colume generated by rotating the region bounded by the graphs of the given equations:<\\\/h3><ul><li><h4>(a) About the x-axis<\\\/h4><\\\/li><\\\/ul><p><img class=\\\"size-medium wp-image-6492 aligncenter\\\" src=\\\"https:\\\/\\\/bisonmath.com\\\/wp-content\\\/uploads\\\/2022\\\/03\\\/Screenshot-25-300x300.png\\\" alt=\\\"\\\" width=\\\"300\\\" height=\\\"300\\\" \\\/><\\\/p><h4>Sketched area to rotate around the axis. Given the airgap between the shaded area and the axis of rotation, and that the chosen differential is perpendicular to the axis, use the washer method. Let \\\\(R = x^{\\\\frac{2}{3}}\\\\) and \\\\(r = x^{\\\\frac{3}{2}}\\\\). Using this data in the integral yields:<\\\/h4>\\\\(<br \\\/>\\\\begin{align*}<br \\\/>&amp; V = \\\\pi \\\\int^1_0 \\\\left[ \\\\left(x^{\\\\frac{2}{3}}\\\\right)^2 \\\\:- \\\\left(x^{\\\\frac{3}{2}}\\\\right)^2 \\\\right] dx = \\\\left[ \\\\frac{3}{7}x^{\\\\frac{7}{3}} \\\\:- \\\\frac{1}{4}x^4 \\\\right] \\\\Bigg\\\\rvert^1_0 = \\\\left[ \\\\frac{3}{7}(1)^{\\\\frac{7}{3}} \\\\:- \\\\frac{1}{4}(1)^4 \\\\right] \\\\:- \\\\left[ \\\\frac{3}{7}(0)^{\\\\frac{7}{3}} \\\\:- \\\\frac{1}{4}(0)^4 \\\\right] \\\\\\\\[5pt]<br \\\/>&amp; \\\\color{green}{V = \\\\frac{5}{28}\\\\pi}<br \\\/>\\\\end{align*}<br \\\/>\\\\)<p>\\u00a0<\\\/p><ul><li><h4>(b) About x=-2<\\\/h4><\\\/li><\\\/ul><p><img class=\\\"size-medium wp-image-6512 aligncenter\\\" src=\\\"https:\\\/\\\/bisonmath.com\\\/wp-content\\\/uploads\\\/2022\\\/03\\\/Screenshot-26-300x300.png\\\" alt=\\\"\\\" width=\\\"300\\\" height=\\\"300\\\" \\\/><\\\/p><h4>Sketched area and updated axis of rotation. Given how we can&#8217;t change the orientation of the differential (mainly because both ends have to touch both functions), we choose to use the shell method. For that, we need to identify the radius and the height. Let \\\\(R = x^{\\\\frac{2}{3}} &#8211; x^{\\\\frac{3}{2}}\\\\) and \\\\(H = x + 2\\\\). Using this values in the integral yields:<\\\/h4>\\\\(<br \\\/>\\\\begin{align*}<br \\\/>&amp; V = 2\\\\pi \\\\int^1_0 (x+2) \\\\left(x^{\\\\frac{2}{3}} &#8211; x^{\\\\frac{3}{2}}\\\\right) dx = 2\\\\pi \\\\int^1_0 \\\\left( x^{\\\\frac{5}{3}} \\\\:- x^{\\\\frac{5}{2}} \\\\:+ 2x^{\\\\frac{2}{3}} \\\\:- 2x^{\\\\frac{3}{2}} \\\\right)dx = 2\\\\pi\\\\left[ \\\\frac{3}{8}x^{\\\\frac{8}{3}} \\\\:- \\\\frac{2}{7}x^{\\\\frac{7}{2}} \\\\:+ (2)\\\\frac{3}{5}x^{\\\\frac{5}{3}} \\\\:- (2)\\\\frac{2}{5}x^{\\\\frac{5}{2}} \\\\right] \\\\Bigg\\\\rvert^1_0 \\\\\\\\[5pt]<br \\\/>&amp; V = 2\\\\pi \\\\left( \\\\left[ \\\\frac{3}{8}(1)^{\\\\frac{8}{3}} \\\\:- \\\\frac{2}{7}(1)^{\\\\frac{7}{2}} \\\\:+ \\\\frac{6}{5}(1)^{\\\\frac{5}{3}} \\\\:- \\\\frac{4}{5}(1)^{\\\\frac{5}{2}} \\\\right] \\\\:- \\\\left[ \\\\frac{3}{8}(0)^{\\\\frac{8}{3}} \\\\:- \\\\frac{2}{7}(0)^{\\\\frac{7}{2}} \\\\:+ \\\\frac{6}{5}(0)^{\\\\frac{5}{3}} \\\\:- \\\\frac{4}{5}(0)^{\\\\frac{5}{2}} \\\\right] \\\\right) \\\\\\\\[5pt]<br \\\/>&amp; V = 2\\\\pi \\\\left[ \\\\frac{3}{8} \\\\:- \\\\frac{2}{7} \\\\:+ \\\\frac{6}{5} \\\\:- \\\\frac{4}{5} \\\\right] = 2\\\\pi \\\\left[\\\\frac{137}{280}\\\\right] \\\\\\\\[5pt]<br \\\/>&amp; \\\\color{green}{V = \\\\frac{137}{140}\\\\pi}<br \\\/>\\\\end{align*}<br \\\/>\\\\)\"}}]}<\/script>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-6fc01187 elementor-widget elementor-widget-toggle\" data-id=\"6fc01187\" data-element_type=\"widget\" data-widget_type=\"toggle.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<div class=\"elementor-toggle\">\n\t\t\t\t\t\t\t<div class=\"elementor-toggle-item\">\n\t\t\t\t\t<h2 id=\"elementor-tab-title-1871\" class=\"elementor-tab-title\" data-tab=\"1\" role=\"button\" aria-controls=\"elementor-tab-content-1871\" aria-expanded=\"false\">\n\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon elementor-toggle-icon-left\" aria-hidden=\"true\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon-closed\"><i class=\"fas fa-toggle-off\"><\/i><\/span>\n\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon-opened\"><i class=\"elementor-toggle-icon-opened fas fa-toggle-on\"><\/i><\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t\t<\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t<a class=\"elementor-toggle-title\" tabindex=\"0\">Problem 5: Solution<\/a>\n\t\t\t\t\t<\/h2>\n\n\t\t\t\t\t<div id=\"elementor-tab-content-1871\" class=\"elementor-tab-content elementor-clearfix\" data-tab=\"1\" role=\"region\" aria-labelledby=\"elementor-tab-title-1871\"><h3>Compute the volume generated by rotating the region bounded by the graphs of the given equations:<\/h3><ul><li><h4>(a) About the y-axis<\/h4><\/li><\/ul><p><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-7008 size-medium aligncenter\" src=\"https:\/\/bisonmath.com\/wp-content\/uploads\/2022\/03\/Screenshot-39-300x300.png\" alt=\"\" width=\"300\" height=\"300\" \/><\/p><h4>Sketched area to be rotated around the y-axis.<br \/>Given the parallel orientation of the differential line regarding the axis of rotation, use the disk method to compute the revolution solid. Calculate the main f(x) function in terms of y: \\(f(y) = sin(y)\\). The limits of integration are \\(0 \\le y \\le 1\\):<\/h4>\\(<br \/>\\begin{align*}<br \/>&amp; V = \\pi \\int^1_0 \\texttt{sin}^2(y) dy = \\frac{\\pi}{2} \\int^1_0 (1 \\:- \\texttt{cos}(2y))dy \\\\[5pt]<br \/>&amp; V = \\frac{\\pi}{2} \\left[ y \\:- \\frac{1}{2}\\texttt{sin}(2y) \\right] \\Bigg\\rvert^1_0 = \\frac{\\pi}{2} \\left[ \\left((1)\u00a0\\:- \\frac{1}{2}\\texttt{sin}(2(1))\\right) \\:- \\left((0) \\:- \\frac{1}{2}\\texttt{sin}(2(0))\\right) \\right] \\\\[5pt]<br \/>&amp; V = \\frac{1}{2} \\left[ 1 \\:- \\frac{1}{2}\\texttt{sin}(2) + \\frac{1}{2}\\texttt{sin}(0) \\right] = \\frac{1}{2} \\left[ 1 \\:- \\texttt{sin}(2) \\right] \\\\[5pt]<br \/>&amp; \\color{green}{V = \\frac{1}{2} \\:- \\frac{1}{2}\\texttt{sin}(2)}<br \/>\\end{align*}<br \/>\\)<p>\u00a0<\/p><ul><li><h4>(b) About x=2<\/h4><\/li><\/ul><p><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-7029 aligncenter\" src=\"https:\/\/bisonmath.com\/wp-content\/uploads\/2022\/03\/Screenshot-40-300x300.png\" alt=\"\" width=\"300\" height=\"300\" \/><\/p><h4>Sketched area to be rotated around \\(x = 2\\) as an axis of revolution.<br \/>The main problem remains almost the same as the last one, but this time we have an airgap in between the rotational area instead.<br \/>Given the orientation of the differential line (as it is perpendicular to the axis of rotation), the best method to use in this case is the washer method.<\/h4>\\(<br \/>\\begin{align*}<br \/>&amp; R = 2; \\Rightarrow R^2 = 4 \\\\[5pt]<br \/>&amp; r = 2 \\:- \\texttt{sin}(y); \\Rightarrow r^2 = (2 \\:- \\texttt{sin}(y))^2 = 4 \\:- 4\\texttt{sin}(y) + \\texttt{sin}^2(y) \\\\[15pt]<br \/>&amp; V = \\pi \\int^1_0 (R^2 \\:- r^2) dy = \\pi \\int^1_0 \\left[ 4 \\:- (4 \\:- 4\\texttt{sin}(y) + \\texttt{sin}^2(y)) \\right] dy \\\\[5pt]<br \/>&amp; V = \\pi \\int^1_0 \\left( 4\\texttt{sin}(y) \\:- \\texttt{sin}^2(y) \\right) dy = \\pi \\left[ 4\\int^1_0 \\texttt{sin}(y)dy \\:- \\int^1_0 \\texttt{sin}^2(y)dy \\right] \\\\[5pt]<br \/>&amp; V = \\pi \\left[ -4\\texttt{cos}(y) \\:- \\frac{1}{2}\\int^1_0 (1 \\:- \\texttt{cos}(2y))dy \\right] = \\pi \\left[ -4\\texttt{cos}(y) \\:- \\frac{1}{2}y + \\frac{1}{4}\\texttt{sin}(2y) \\right] \\Bigg\\rvert^1_0 \\\\[5pt]<br \/>&amp; V = \\pi \\left( \\left[ -4\\texttt{cos}(1) \\:- \\frac{1}{2}(1) + \\frac{1}{4}\\texttt{sin}(2(1)) \\right] \\:- \\left[ -4\\texttt{cos}(0) \\:- \\frac{1}{2}(0) + \\frac{1}{4}\\texttt{sin}(2(0)) \\right] \\right) \\\\[5pt]<br \/>&amp; V = \\pi \\left[ -4\\texttt{cos}(1) \\:- \\frac{1}{2} + \\frac{1}{4}\\texttt{sin}(2) + 4 \\right] \\\\[5pt]<br \/>&amp; \\color{green}{V = \\pi \\left[ \\frac{1}{4}\\texttt{sin}(2) \\:- 4\\texttt{cos}(1) + \\frac{7}{2} \\right]}<br \/>\\end{align*}<br \/>\\)<p>\u00a0<\/p><\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t\t<script type=\"application\/ld+json\">{\"@context\":\"https:\\\/\\\/schema.org\",\"@type\":\"FAQPage\",\"mainEntity\":[{\"@type\":\"Question\",\"name\":\"Problem 5: Solution\",\"acceptedAnswer\":{\"@type\":\"Answer\",\"text\":\"<h3>Compute the volume generated by rotating the region bounded by the graphs of the given equations:<\\\/h3><ul><li><h4>(a) About the y-axis<\\\/h4><\\\/li><\\\/ul><p><img class=\\\"wp-image-7008 size-medium aligncenter\\\" src=\\\"https:\\\/\\\/bisonmath.com\\\/wp-content\\\/uploads\\\/2022\\\/03\\\/Screenshot-39-300x300.png\\\" alt=\\\"\\\" width=\\\"300\\\" height=\\\"300\\\" \\\/><\\\/p><h4>Sketched area to be rotated around the y-axis.<br \\\/>Given the parallel orientation of the differential line regarding the axis of rotation, use the disk method to compute the revolution solid. Calculate the main f(x) function in terms of y: \\\\(f(y) = sin(y)\\\\). The limits of integration are \\\\(0 \\\\le y \\\\le 1\\\\):<\\\/h4>\\\\(<br \\\/>\\\\begin{align*}<br \\\/>&amp; V = \\\\pi \\\\int^1_0 \\\\texttt{sin}^2(y) dy = \\\\frac{\\\\pi}{2} \\\\int^1_0 (1 \\\\:- \\\\texttt{cos}(2y))dy \\\\\\\\[5pt]<br \\\/>&amp; V = \\\\frac{\\\\pi}{2} \\\\left[ y \\\\:- \\\\frac{1}{2}\\\\texttt{sin}(2y) \\\\right] \\\\Bigg\\\\rvert^1_0 = \\\\frac{\\\\pi}{2} \\\\left[ \\\\left((1)\\u00a0\\\\:- \\\\frac{1}{2}\\\\texttt{sin}(2(1))\\\\right) \\\\:- \\\\left((0) \\\\:- \\\\frac{1}{2}\\\\texttt{sin}(2(0))\\\\right) \\\\right] \\\\\\\\[5pt]<br \\\/>&amp; V = \\\\frac{1}{2} \\\\left[ 1 \\\\:- \\\\frac{1}{2}\\\\texttt{sin}(2) + \\\\frac{1}{2}\\\\texttt{sin}(0) \\\\right] = \\\\frac{1}{2} \\\\left[ 1 \\\\:- \\\\texttt{sin}(2) \\\\right] \\\\\\\\[5pt]<br \\\/>&amp; \\\\color{green}{V = \\\\frac{1}{2} \\\\:- \\\\frac{1}{2}\\\\texttt{sin}(2)}<br \\\/>\\\\end{align*}<br \\\/>\\\\)<p>\\u00a0<\\\/p><ul><li><h4>(b) About x=2<\\\/h4><\\\/li><\\\/ul><p><img class=\\\"size-medium wp-image-7029 aligncenter\\\" src=\\\"https:\\\/\\\/bisonmath.com\\\/wp-content\\\/uploads\\\/2022\\\/03\\\/Screenshot-40-300x300.png\\\" alt=\\\"\\\" width=\\\"300\\\" height=\\\"300\\\" \\\/><\\\/p><h4>Sketched area to be rotated around \\\\(x = 2\\\\) as an axis of revolution.<br \\\/>The main problem remains almost the same as the last one, but this time we have an airgap in between the rotational area instead.<br \\\/>Given the orientation of the differential line (as it is perpendicular to the axis of rotation), the best method to use in this case is the washer method.<\\\/h4>\\\\(<br \\\/>\\\\begin{align*}<br \\\/>&amp; R = 2; \\\\Rightarrow R^2 = 4 \\\\\\\\[5pt]<br \\\/>&amp; r = 2 \\\\:- \\\\texttt{sin}(y); \\\\Rightarrow r^2 = (2 \\\\:- \\\\texttt{sin}(y))^2 = 4 \\\\:- 4\\\\texttt{sin}(y) + \\\\texttt{sin}^2(y) \\\\\\\\[15pt]<br \\\/>&amp; V = \\\\pi \\\\int^1_0 (R^2 \\\\:- r^2) dy = \\\\pi \\\\int^1_0 \\\\left[ 4 \\\\:- (4 \\\\:- 4\\\\texttt{sin}(y) + \\\\texttt{sin}^2(y)) \\\\right] dy \\\\\\\\[5pt]<br \\\/>&amp; V = \\\\pi \\\\int^1_0 \\\\left( 4\\\\texttt{sin}(y) \\\\:- \\\\texttt{sin}^2(y) \\\\right) dy = \\\\pi \\\\left[ 4\\\\int^1_0 \\\\texttt{sin}(y)dy \\\\:- \\\\int^1_0 \\\\texttt{sin}^2(y)dy \\\\right] \\\\\\\\[5pt]<br \\\/>&amp; V = \\\\pi \\\\left[ -4\\\\texttt{cos}(y) \\\\:- \\\\frac{1}{2}\\\\int^1_0 (1 \\\\:- \\\\texttt{cos}(2y))dy \\\\right] = \\\\pi \\\\left[ -4\\\\texttt{cos}(y) \\\\:- \\\\frac{1}{2}y + \\\\frac{1}{4}\\\\texttt{sin}(2y) \\\\right] \\\\Bigg\\\\rvert^1_0 \\\\\\\\[5pt]<br \\\/>&amp; V = \\\\pi \\\\left( \\\\left[ -4\\\\texttt{cos}(1) \\\\:- \\\\frac{1}{2}(1) + \\\\frac{1}{4}\\\\texttt{sin}(2(1)) \\\\right] \\\\:- \\\\left[ -4\\\\texttt{cos}(0) \\\\:- \\\\frac{1}{2}(0) + \\\\frac{1}{4}\\\\texttt{sin}(2(0)) \\\\right] \\\\right) \\\\\\\\[5pt]<br \\\/>&amp; V = \\\\pi \\\\left[ -4\\\\texttt{cos}(1) \\\\:- \\\\frac{1}{2} + \\\\frac{1}{4}\\\\texttt{sin}(2) + 4 \\\\right] \\\\\\\\[5pt]<br \\\/>&amp; \\\\color{green}{V = \\\\pi \\\\left[ \\\\frac{1}{4}\\\\texttt{sin}(2) \\\\:- 4\\\\texttt{cos}(1) + \\\\frac{7}{2} \\\\right]}<br \\\/>\\\\end{align*}<br \\\/>\\\\)<p>\\u00a0<\\\/p>\"}}]}<\/script>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-21d1b5ca elementor-widget elementor-widget-toggle\" data-id=\"21d1b5ca\" data-element_type=\"widget\" data-widget_type=\"toggle.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<div class=\"elementor-toggle\">\n\t\t\t\t\t\t\t<div class=\"elementor-toggle-item\">\n\t\t\t\t\t<h2 id=\"elementor-tab-title-5671\" class=\"elementor-tab-title\" data-tab=\"1\" role=\"button\" aria-controls=\"elementor-tab-content-5671\" aria-expanded=\"false\">\n\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon elementor-toggle-icon-left\" aria-hidden=\"true\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon-closed\"><i class=\"fas fa-toggle-off\"><\/i><\/span>\n\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon-opened\"><i class=\"elementor-toggle-icon-opened fas fa-toggle-on\"><\/i><\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t\t<\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t<a class=\"elementor-toggle-title\" tabindex=\"0\">Problem 6: Solution<\/a>\n\t\t\t\t\t<\/h2>\n\n\t\t\t\t\t<div id=\"elementor-tab-content-5671\" class=\"elementor-tab-content elementor-clearfix\" data-tab=\"1\" role=\"region\" aria-labelledby=\"elementor-tab-title-5671\"><h3>Prove that by rotating the circle \\((x-a)^2 + y^2 = r^2\\) about the y-axis generates a torus whose volume is \\(V = 2\\pi^2ar^2\\):<\/h3><p><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-6554 aligncenter\" src=\"https:\/\/bisonmath.com\/wp-content\/uploads\/2022\/03\/Screenshot-27-300x300.png\" alt=\"\" width=\"300\" height=\"300\" \/><\/p><h4>Sketched area to be rotated around the y-axis. For the sake of an easier integration process, let everything be with respect to y. The initial equation of the circle would be:<\/h4>\\(<br \/>\\begin{align*}<br \/>(x-a)^2 + y^2 = r^2 \\\\[5pt]<br \/>(x-a)^2 = r^2 &#8211; y^2 \\\\[5pt]<br \/>x-a = \\sqrt{r^2-y^2} \\\\[5pt]<br \/>x = \\sqrt{r^2 &#8211; y^2} + a<br \/>\\end{align*}<br \/>\\)<p>\u00a0<\/p><h4>Given the function is a circle, the integration limits with regards to y are -r and r. We will use the washer method in order to calculate the volume of the shape:<\/h4>\\(<br \/>\\begin{align*}<br \/>&amp; R = \\sqrt{r^2 &#8211; y^2} + a; \\Rightarrow R^2 = \\left(a+\\sqrt{r^2 &#8211; y^2}\\right)^2 = a^2 + 2a\\sqrt{r^2 &#8211; y^2} + r^2-y^2 \\\\[5pt]<br \/>&amp; r = -\\sqrt{r^2 &#8211; y^2} + a; \\Rightarrow R^2 = \\left(a-\\sqrt{r^2 &#8211; y^2}\\right)^2 =\u00a0a^2 &#8211; 2a\\sqrt{r^2 &#8211; y^2} + r^2-y^2 \\\\[5pt]<br \/>&amp; V = \\pi \\int^r_{-r} \\left(R^2 &#8211; r^2\\right) dy = \\pi \\int^r_{-r} \\left(\\left[a^2 + 2a\\sqrt{r^2 &#8211; y^2} + r^2-y^2\\right] &#8211; \\left[ a^2 &#8211; 2a\\sqrt{r^2 &#8211; y^2} + r^2-y^2 \\right] \\right) dy \\\\[5pt]<br \/>&amp; V = \\pi \\int^r_{-r} \\left(a^2 + 2a\\sqrt{r^2 &#8211; y^2} + r^2 \\:- y^2 \\:-\u00a0 a^2 + 2a\\sqrt{r^2 &#8211; y^2} \\:- r^2 \\:+ y^2 \\right) dy = \\pi \\int^r_{-r} 4a\\sqrt{r^2-y^2} dy \\\\[5pt]<br \/>&amp; V = 4a\\pi \\int^r_{-r} \\sqrt{r^2 \\:- y^2} dy = 8a\\pi \\int^r_0 \\sqrt{r^2 \\:- y^2} dy<br \/>\\end{align*}<br \/>\\)<p>\u00a0<\/p><h4>Apply trigonometric substitution in order to solve the final integral. Let \\(y = r\\texttt{sin}(\\theta)\\), so \\(dy = r\\texttt{cos}(\\theta)d\\theta\\). Applying this subsitution in the radical yields:<\/h4>\\(<br \/>\\begin{align*}<br \/>&amp; \\sqrt{r^2 \\:- y^2} = \\sqrt{r^2 \\:- (r\\texttt{sin}(\\theta))^2} = \\sqrt{r^2 \\:- r^2\\texttt{sin}^2(\\theta)} = \\sqrt{r^2 (1 &#8211; \\texttt{sin}^2(\\theta))} = \\sqrt{r^2\\texttt{cos}^2(\\theta)} = r\\texttt{cos}(\\theta) \\\\[5pt]<br \/>&amp; V = 8a\\pi \\int^r_0 \\sqrt{r^2 \\:- y^2} = 8a\\pi \\int^r_0 \\left[r\\texttt{cos}(\\theta)\\right] \\left[r\\texttt{cos}(\\theta)d\\theta\\right] = 8ar^2\\pi \\int^r_0 \\texttt{cos}^2(\\theta)d\\theta = 8ar^2\\pi \\left(\\frac{1}{2}\\right)\\int^r_0 \\left(1 + \\texttt{cos}(2\\theta)\\right) d\\theta \\\\[5pt]<br \/>&amp; V = 4ar^2\\pi \\left[ \\theta \\:+ \\frac{1}{2}\\texttt{sin}(2\\theta) \\right] \\Bigg\\rvert^r_0 = 4ar^2\\pi \\left[ \\theta \\:+ \\texttt{sin}(\\theta)\\texttt{cos}(\\theta) \\right] \\Bigg\\rvert^r_0 \\\\[5pt]<br \/>\\end{align*}<br \/>\\)<p>\u00a0<\/p><h4>Substitute back to the original variable by using the \\(\\texttt{sin}(\\theta) = \\frac{y}{r}\\) substitution done earlier:<\/h4>\\(<br \/>\\begin{align*}<br \/>&amp; V = 4ar^2\\pi \\left[ \\theta \\:+ \\texttt{sin}(\\theta)\\texttt{cos}(\\theta) \\right] \\Bigg\\rvert^r_0 = 4ar^2 \\pi \\left[ \\texttt{Arcsin}\\left( \\frac{y}{r} \\right) \\:+ \\left(\\frac{y}{r}\\right)\\left(\\frac{\\sqrt{r^2-y^2}}{r}\\right)\\right] \\Bigg\\rvert^r_0 \\\\[5pt]<br \/>&amp; V = 4ar^2 \\pi \\left[ \\texttt{Arcsin}\\left( \\frac{y}{r} \\right) \\:+ \\frac{y\\sqrt{r^2-y^2}}{r^2} \\right] \\Bigg\\rvert^r_0 \\\\[5pt]<br \/>&amp; V = 4ar^2 \\pi \\left[ \\left( \\texttt{Arcsin}\\left( \\frac{r}{r} \\right) \\:+ \\frac{r\\sqrt{r^2-r^2}}{r^2}\\right) \\:- \\left( \\texttt{Arcsin}\\left( \\frac{0}{r} \\right) \\:+ \\frac{0\\sqrt{r^2-0^2}}{r^2}\\right) \\right] = 4ar^2 \\pi \\left[ \\texttt{Arcsin}(1) \\:+ 0 \\:- \\texttt{Arcsin}(0) \\:- 0 \\right] \\\\[5pt]<br \/>&amp; V = 4ar^2\\pi \\left[ \\frac{\\pi}{2} \\:- 0 \\right] = 4ar^2\\pi\\left[ \\frac{\\pi}{2}\\right] \\\\[5pt]<br \/>&amp; \\color{green}{V = 2ar^2\\pi^2}<br \/>\\end{align*}<br \/>\\)<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t\t<script type=\"application\/ld+json\">{\"@context\":\"https:\\\/\\\/schema.org\",\"@type\":\"FAQPage\",\"mainEntity\":[{\"@type\":\"Question\",\"name\":\"Problem 6: Solution\",\"acceptedAnswer\":{\"@type\":\"Answer\",\"text\":\"<h3>Prove that by rotating the circle \\\\((x-a)^2 + y^2 = r^2\\\\) about the y-axis generates a torus whose volume is \\\\(V = 2\\\\pi^2ar^2\\\\):<\\\/h3><p><img class=\\\"size-medium wp-image-6554 aligncenter\\\" src=\\\"https:\\\/\\\/bisonmath.com\\\/wp-content\\\/uploads\\\/2022\\\/03\\\/Screenshot-27-300x300.png\\\" alt=\\\"\\\" width=\\\"300\\\" height=\\\"300\\\" \\\/><\\\/p><h4>Sketched area to be rotated around the y-axis. For the sake of an easier integration process, let everything be with respect to y. The initial equation of the circle would be:<\\\/h4>\\\\(<br \\\/>\\\\begin{align*}<br \\\/>(x-a)^2 + y^2 = r^2 \\\\\\\\[5pt]<br \\\/>(x-a)^2 = r^2 &#8211; y^2 \\\\\\\\[5pt]<br \\\/>x-a = \\\\sqrt{r^2-y^2} \\\\\\\\[5pt]<br \\\/>x = \\\\sqrt{r^2 &#8211; y^2} + a<br \\\/>\\\\end{align*}<br \\\/>\\\\)<p>\\u00a0<\\\/p><h4>Given the function is a circle, the integration limits with regards to y are -r and r. We will use the washer method in order to calculate the volume of the shape:<\\\/h4>\\\\(<br \\\/>\\\\begin{align*}<br \\\/>&amp; R = \\\\sqrt{r^2 &#8211; y^2} + a; \\\\Rightarrow R^2 = \\\\left(a+\\\\sqrt{r^2 &#8211; y^2}\\\\right)^2 = a^2 + 2a\\\\sqrt{r^2 &#8211; y^2} + r^2-y^2 \\\\\\\\[5pt]<br \\\/>&amp; r = -\\\\sqrt{r^2 &#8211; y^2} + a; \\\\Rightarrow R^2 = \\\\left(a-\\\\sqrt{r^2 &#8211; y^2}\\\\right)^2 =\\u00a0a^2 &#8211; 2a\\\\sqrt{r^2 &#8211; y^2} + r^2-y^2 \\\\\\\\[5pt]<br \\\/>&amp; V = \\\\pi \\\\int^r_{-r} \\\\left(R^2 &#8211; r^2\\\\right) dy = \\\\pi \\\\int^r_{-r} \\\\left(\\\\left[a^2 + 2a\\\\sqrt{r^2 &#8211; y^2} + r^2-y^2\\\\right] &#8211; \\\\left[ a^2 &#8211; 2a\\\\sqrt{r^2 &#8211; y^2} + r^2-y^2 \\\\right] \\\\right) dy \\\\\\\\[5pt]<br \\\/>&amp; V = \\\\pi \\\\int^r_{-r} \\\\left(a^2 + 2a\\\\sqrt{r^2 &#8211; y^2} + r^2 \\\\:- y^2 \\\\:-\\u00a0 a^2 + 2a\\\\sqrt{r^2 &#8211; y^2} \\\\:- r^2 \\\\:+ y^2 \\\\right) dy = \\\\pi \\\\int^r_{-r} 4a\\\\sqrt{r^2-y^2} dy \\\\\\\\[5pt]<br \\\/>&amp; V = 4a\\\\pi \\\\int^r_{-r} \\\\sqrt{r^2 \\\\:- y^2} dy = 8a\\\\pi \\\\int^r_0 \\\\sqrt{r^2 \\\\:- y^2} dy<br \\\/>\\\\end{align*}<br \\\/>\\\\)<p>\\u00a0<\\\/p><h4>Apply trigonometric substitution in order to solve the final integral. Let \\\\(y = r\\\\texttt{sin}(\\\\theta)\\\\), so \\\\(dy = r\\\\texttt{cos}(\\\\theta)d\\\\theta\\\\). Applying this subsitution in the radical yields:<\\\/h4>\\\\(<br \\\/>\\\\begin{align*}<br \\\/>&amp; \\\\sqrt{r^2 \\\\:- y^2} = \\\\sqrt{r^2 \\\\:- (r\\\\texttt{sin}(\\\\theta))^2} = \\\\sqrt{r^2 \\\\:- r^2\\\\texttt{sin}^2(\\\\theta)} = \\\\sqrt{r^2 (1 &#8211; \\\\texttt{sin}^2(\\\\theta))} = \\\\sqrt{r^2\\\\texttt{cos}^2(\\\\theta)} = r\\\\texttt{cos}(\\\\theta) \\\\\\\\[5pt]<br \\\/>&amp; V = 8a\\\\pi \\\\int^r_0 \\\\sqrt{r^2 \\\\:- y^2} = 8a\\\\pi \\\\int^r_0 \\\\left[r\\\\texttt{cos}(\\\\theta)\\\\right] \\\\left[r\\\\texttt{cos}(\\\\theta)d\\\\theta\\\\right] = 8ar^2\\\\pi \\\\int^r_0 \\\\texttt{cos}^2(\\\\theta)d\\\\theta = 8ar^2\\\\pi \\\\left(\\\\frac{1}{2}\\\\right)\\\\int^r_0 \\\\left(1 + \\\\texttt{cos}(2\\\\theta)\\\\right) d\\\\theta \\\\\\\\[5pt]<br \\\/>&amp; V = 4ar^2\\\\pi \\\\left[ \\\\theta \\\\:+ \\\\frac{1}{2}\\\\texttt{sin}(2\\\\theta) \\\\right] \\\\Bigg\\\\rvert^r_0 = 4ar^2\\\\pi \\\\left[ \\\\theta \\\\:+ \\\\texttt{sin}(\\\\theta)\\\\texttt{cos}(\\\\theta) \\\\right] \\\\Bigg\\\\rvert^r_0 \\\\\\\\[5pt]<br \\\/>\\\\end{align*}<br \\\/>\\\\)<p>\\u00a0<\\\/p><h4>Substitute back to the original variable by using the \\\\(\\\\texttt{sin}(\\\\theta) = \\\\frac{y}{r}\\\\) substitution done earlier:<\\\/h4>\\\\(<br \\\/>\\\\begin{align*}<br \\\/>&amp; V = 4ar^2\\\\pi \\\\left[ \\\\theta \\\\:+ \\\\texttt{sin}(\\\\theta)\\\\texttt{cos}(\\\\theta) \\\\right] \\\\Bigg\\\\rvert^r_0 = 4ar^2 \\\\pi \\\\left[ \\\\texttt{Arcsin}\\\\left( \\\\frac{y}{r} \\\\right) \\\\:+ \\\\left(\\\\frac{y}{r}\\\\right)\\\\left(\\\\frac{\\\\sqrt{r^2-y^2}}{r}\\\\right)\\\\right] \\\\Bigg\\\\rvert^r_0 \\\\\\\\[5pt]<br \\\/>&amp; V = 4ar^2 \\\\pi \\\\left[ \\\\texttt{Arcsin}\\\\left( \\\\frac{y}{r} \\\\right) \\\\:+ \\\\frac{y\\\\sqrt{r^2-y^2}}{r^2} \\\\right] \\\\Bigg\\\\rvert^r_0 \\\\\\\\[5pt]<br \\\/>&amp; V = 4ar^2 \\\\pi \\\\left[ \\\\left( \\\\texttt{Arcsin}\\\\left( \\\\frac{r}{r} \\\\right) \\\\:+ \\\\frac{r\\\\sqrt{r^2-r^2}}{r^2}\\\\right) \\\\:- \\\\left( \\\\texttt{Arcsin}\\\\left( \\\\frac{0}{r} \\\\right) \\\\:+ \\\\frac{0\\\\sqrt{r^2-0^2}}{r^2}\\\\right) \\\\right] = 4ar^2 \\\\pi \\\\left[ \\\\texttt{Arcsin}(1) \\\\:+ 0 \\\\:- \\\\texttt{Arcsin}(0) \\\\:- 0 \\\\right] \\\\\\\\[5pt]<br \\\/>&amp; V = 4ar^2\\\\pi \\\\left[ \\\\frac{\\\\pi}{2} \\\\:- 0 \\\\right] = 4ar^2\\\\pi\\\\left[ \\\\frac{\\\\pi}{2}\\\\right] \\\\\\\\[5pt]<br \\\/>&amp; \\\\color{green}{V = 2ar^2\\\\pi^2}<br \\\/>\\\\end{align*}<br \\\/>\\\\)\"}}]}<\/script>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-ad24a7e elementor-widget elementor-widget-toggle\" data-id=\"ad24a7e\" data-element_type=\"widget\" data-widget_type=\"toggle.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<div class=\"elementor-toggle\">\n\t\t\t\t\t\t\t<div class=\"elementor-toggle-item\">\n\t\t\t\t\t<h2 id=\"elementor-tab-title-1811\" class=\"elementor-tab-title\" data-tab=\"1\" role=\"button\" aria-controls=\"elementor-tab-content-1811\" aria-expanded=\"false\">\n\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon elementor-toggle-icon-left\" aria-hidden=\"true\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon-closed\"><i class=\"fas fa-toggle-off\"><\/i><\/span>\n\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon-opened\"><i class=\"elementor-toggle-icon-opened fas fa-toggle-on\"><\/i><\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t\t<\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t<a class=\"elementor-toggle-title\" tabindex=\"0\">Problem 7: Solution<\/a>\n\t\t\t\t\t<\/h2>\n\n\t\t\t\t\t<div id=\"elementor-tab-content-1811\" class=\"elementor-tab-content elementor-clearfix\" data-tab=\"1\" role=\"region\" aria-labelledby=\"elementor-tab-title-1811\"><h3>Compute the volume generated by rotating the region bounded by the graphs of the given equation about the y-axis:<\/h3><p><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-6663 aligncenter\" src=\"https:\/\/bisonmath.com\/wp-content\/uploads\/2022\/03\/Screenshot-28-300x300.png\" alt=\"\" width=\"300\" height=\"300\" \/><\/p><h3>Sketched area to be rotated. Given how the differential is located parallel to the axis of rotation, use the shell method. Let \\(R = x \\) and \\(H = 4 \\:- x^2 \\:- (2 \\:- x) = 2 + x \\:- x^2\\). Using the aforementioned values into the integrand yields:<\/h3>\\(<br \/>\\begin{align*}<br \/>&amp; V = 2\\pi \\int^2_0 x(2 + x \\:- x^2) dx = 2\\pi \\int^2_0 (2x + x^2 \\:- x^3)dx = 2\\pi \\left[ \\frac{1}{2}2x^2 + \\frac{1}{3}x^3 \\:- \\frac{1}{4}x^4 \\right] \\Bigg\\rvert^2_0 = 2\\pi \\left[ x^2 + \\frac{1}{3}x^3 \\:- \\frac{1}{4}x^4 \\right] \\Bigg\\rvert^2_0 \\\\[5pt]<br \/>&amp; V = 2\\pi \\left( \\left[ (2)^2 + \\frac{1}{3}(2)^3 \\:- \\frac{1}{4}(2)^4 \\right] \\:- \\left[ (0)^2 + \\frac{1}{3}(0)^3 \\:- \\frac{1}{4}(0)^4 \\right] \\right) = 2\\pi \\left[4 + \\frac{8}{3} \\:- 4\\right] = 2\\pi \\left[\\frac{8}{3}\\right] \\\\[5pt]<br \/>&amp; \\color{green}{V = \\frac{16}{3}\\pi}<br \/>\\end{align*}<br \/>\\)<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t\t<script type=\"application\/ld+json\">{\"@context\":\"https:\\\/\\\/schema.org\",\"@type\":\"FAQPage\",\"mainEntity\":[{\"@type\":\"Question\",\"name\":\"Problem 7: Solution\",\"acceptedAnswer\":{\"@type\":\"Answer\",\"text\":\"<h3>Compute the volume generated by rotating the region bounded by the graphs of the given equation about the y-axis:<\\\/h3><p><img class=\\\"size-medium wp-image-6663 aligncenter\\\" src=\\\"https:\\\/\\\/bisonmath.com\\\/wp-content\\\/uploads\\\/2022\\\/03\\\/Screenshot-28-300x300.png\\\" alt=\\\"\\\" width=\\\"300\\\" height=\\\"300\\\" \\\/><\\\/p><h3>Sketched area to be rotated. Given how the differential is located parallel to the axis of rotation, use the shell method. Let \\\\(R = x \\\\) and \\\\(H = 4 \\\\:- x^2 \\\\:- (2 \\\\:- x) = 2 + x \\\\:- x^2\\\\). Using the aforementioned values into the integrand yields:<\\\/h3>\\\\(<br \\\/>\\\\begin{align*}<br \\\/>&amp; V = 2\\\\pi \\\\int^2_0 x(2 + x \\\\:- x^2) dx = 2\\\\pi \\\\int^2_0 (2x + x^2 \\\\:- x^3)dx = 2\\\\pi \\\\left[ \\\\frac{1}{2}2x^2 + \\\\frac{1}{3}x^3 \\\\:- \\\\frac{1}{4}x^4 \\\\right] \\\\Bigg\\\\rvert^2_0 = 2\\\\pi \\\\left[ x^2 + \\\\frac{1}{3}x^3 \\\\:- \\\\frac{1}{4}x^4 \\\\right] \\\\Bigg\\\\rvert^2_0 \\\\\\\\[5pt]<br \\\/>&amp; V = 2\\\\pi \\\\left( \\\\left[ (2)^2 + \\\\frac{1}{3}(2)^3 \\\\:- \\\\frac{1}{4}(2)^4 \\\\right] \\\\:- \\\\left[ (0)^2 + \\\\frac{1}{3}(0)^3 \\\\:- \\\\frac{1}{4}(0)^4 \\\\right] \\\\right) = 2\\\\pi \\\\left[4 + \\\\frac{8}{3} \\\\:- 4\\\\right] = 2\\\\pi \\\\left[\\\\frac{8}{3}\\\\right] \\\\\\\\[5pt]<br \\\/>&amp; \\\\color{green}{V = \\\\frac{16}{3}\\\\pi}<br \\\/>\\\\end{align*}<br \\\/>\\\\)\"}}]}<\/script>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-1b850fd0 elementor-widget elementor-widget-toggle\" data-id=\"1b850fd0\" data-element_type=\"widget\" data-widget_type=\"toggle.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<div class=\"elementor-toggle\">\n\t\t\t\t\t\t\t<div class=\"elementor-toggle-item\">\n\t\t\t\t\t<h2 id=\"elementor-tab-title-4611\" class=\"elementor-tab-title\" data-tab=\"1\" role=\"button\" aria-controls=\"elementor-tab-content-4611\" aria-expanded=\"false\">\n\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon elementor-toggle-icon-left\" aria-hidden=\"true\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon-closed\"><i class=\"fas fa-toggle-off\"><\/i><\/span>\n\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon-opened\"><i class=\"elementor-toggle-icon-opened fas fa-toggle-on\"><\/i><\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t\t<\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t<a class=\"elementor-toggle-title\" tabindex=\"0\">Problem 8: Solution<\/a>\n\t\t\t\t\t<\/h2>\n\n\t\t\t\t\t<div id=\"elementor-tab-content-4611\" class=\"elementor-tab-content elementor-clearfix\" data-tab=\"1\" role=\"region\" aria-labelledby=\"elementor-tab-title-4611\"><h3>Compute the volume generated by rotating the region bounded by the graphs of \\(f(x)=x^3, g(x)=x \\le 0 \\le 1 \\):<\/h3><ul><li><h3>About x = 1<\/h3><\/li><\/ul><p><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-7108 size-medium aligncenter\" src=\"https:\/\/bisonmath.com\/wp-content\/uploads\/2022\/03\/Screenshot-43-300x300.png\" alt=\"\" width=\"300\" height=\"300\" \/><\/p><h4>Use the shell method because of the orientation of the differential line. Let \\(R = 1-x \\) and \\(x \\:- x^3\\). Updating these values in the integral yields:<\/h4>\\(<br \/>\\begin{align*}<br \/>&amp; V = \\pi \\int^1_0 (1-x)(x \\:- x^3)dx = \\pi \\int^1_0 (x^4 \\:- x^3 \\:- x^2 + x)dx \\\\[5pt]<br \/>&amp; V = \\pi \\left[ \\frac{1}{5}x^5 \\:- \\frac{1}{4}x^4 \\:- \\frac{1}{3}x^3 + \\frac{1}{2}x^2 \\right] \\Bigg\\rvert^1_0 \\\\[5pt]<br \/>&amp; V = \\pi \\left( \\left[ \\frac{1}{5}(1)^5 \\:- \\frac{1}{4}(1)^4 \\:- \\frac{1}{3}(1)^3 + \\frac{1}{2}(1)^2 \\right] \\:- \\left[ \\frac{1}{5}(0)^5 \\:- \\frac{1}{4}(0)^4 \\:- \\frac{1}{3}(0)^3 + \\frac{1}{2}(0)^2 \\right] \\right) \\\\[5pt]<br \/>&amp; V = \\pi \\left[ \\frac{1}{5} \\:- \\frac{1}{4} \\:- \\frac{1}{3} + \\frac{1}{2} \\right] \\\\[5pt]<br \/>&amp; \\color{green}{V = \\frac{7}{60}\\pi}<br \/>\\end{align*}<br \/>\\)<p>\u00a0<\/p><ul><li><h3>About x = -2<\/h3><\/li><\/ul><p><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-7112 aligncenter\" src=\"https:\/\/bisonmath.com\/wp-content\/uploads\/2022\/03\/Screenshot-44-300x300.png\" alt=\"\" width=\"300\" height=\"300\" \/><\/p><h4>Once again, use the shell method. The problem is very similar to the last one. Let \\(R = x+2 \\) and \\(x \\:- x^3\\). Updating these values in the integral yields:<\/h4>\\(<br \/>\\begin{align*}<br \/>&amp; V = \\pi \\int^1_0 (x+2)(x \\:- x^3)dx = \\pi \\int^1_0 (-x^4 \\:- 2x^3 + x^2 + 2x)dx \\\\[5pt]<br \/>&amp; V = \\pi \\left[ -\\frac{1}{5}x^5 \\:- \\frac{2}{4}x^4 + \\frac{1}{3}x^3 + \\frac{2}{2}x^2 \\right] \\Bigg\\rvert^1_0\u00a0 = \\pi \\left[ -\\frac{1}{5}x^5 \\:- \\frac{1}{2}x^4 + \\frac{1}{3}x^3 + x^2 \\right] \\Bigg\\rvert^1_0 \\\\[5pt]<br \/>&amp; V = \\pi \\left( \\left[ -\\frac{1}{5}(1)^5 \\:- \\frac{1}{2}(1)^4 + \\frac{1}{3}(1)^3 + (1)^2 \\right] \\:- \\left[ -\\frac{1}{5}(0)^5 \\:- \\frac{1}{2}(0)^4 + \\frac{1}{3}(0)^3 + (0)^2 \\right] \\right) \\\\[5pt]<br \/>&amp; V = \\pi \\left[ -\\frac{1}{5} \\:- \\frac{1}{2} + \\frac{1}{3} + 1 \\right] \\\\[5pt]<br \/>&amp; \\color{green}{V = \\frac{19}{30}\\pi}<br \/>\\end{align*}<br \/>\\)<p>\u00a0<\/p><ul><li><h3>About y = 2<\/h3><\/li><\/ul><p><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-7132 aligncenter\" src=\"https:\/\/bisonmath.com\/wp-content\/uploads\/2022\/03\/Screenshot-45-300x300.png\" alt=\"\" width=\"300\" height=\"300\" \/><\/p><h4>Given the orientation of the differential line with respect to the axis of rotation, use the washer method:<\/h4>\\(<br \/>\\begin{align*}<br \/>&amp; R = 2 \\:- x^3; \\Rightarrow R^2 = (2 \\:- x^3)^2 = 4 \\:- 4x^3 + x^6 \\\\[5pt]<br \/>&amp; r = 2 \\:- x; \\Rightarrow r^2 = (2 \\:- x)^2 = 4 \\:- 4x + x^2 \\\\[10pt]<br \/>&amp; V = \\pi \\int^1_0 (R^2 \\:- r^2)dx = \\pi \\int^1_0 (4 \\:- 4x^3 + x^6 \\:- (4 \\:- 4x + x^2)) dx = \\pi \\int^1_0 (x^6 \\:- 4x^3 \\:- x^2 + 4x)dx \\\\[5pt]<br \/>&amp; V = \\pi \\left[ \\frac{1}{7}x^7 \\:- \\frac{4}{4}x^4 \\:- \\frac{1}{3}x^3 + \\frac{4}{2}x^2 \\right] \\Bigg\\rvert^1_0 = \\pi \\left[ \\frac{1}{7}x^7 \\:- x^4 \\:- \\frac{1}{3}x^3 + 2x^2 \\right] \\Bigg\\rvert^1_0 \\\\[5pt]<br \/>&amp; V = \\pi \\left( \\left[ \\frac{1}{7}(1)^7 \\:- (1)^4 \\:- \\frac{1}{3}(1)^3 + 2(1)^2 \\right] \\:- \\left[ \\frac{1}{7}(0)^7 \\:- (0)^4 \\:- \\frac{1}{3}(0)^3 + 2(0)^2 \\right] \\right) \\\\[5pt]<br \/>&amp; V = \\pi \\left[ \\frac{1}{7} \\:- 1 \\:- \\frac{1}{3} + 2 \\right] \\\\[5pt]<br \/>&amp; \\color{green}{V = \\frac{17}{21}\\pi}<br \/>\\end{align*}<br \/>\\)<p>\u00a0<\/p><ul><li><h3>About y = -2<\/h3><\/li><\/ul><p><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-7171 aligncenter\" src=\"https:\/\/bisonmath.com\/wp-content\/uploads\/2022\/03\/Screenshot-46-300x300.png\" alt=\"\" width=\"300\" height=\"300\" \/><\/p><h4>Once again, apply the washer method because of the orientation of the differential line:<\/h4>\\(<br \/>\\begin{align*}<br \/>&amp; R = x + 2; \\Rightarrow R^2 = (x + 2)^2 = x^2 + 4x + 4 \\\\[5pt]<br \/>&amp; r = x^3 + 2; \\Rightarrow r^2 = (x^3 + 2)^2 = x^6 + 4x^3 + 4 \\\\[10pt]<br \/>&amp; V = \\pi \\int^1_0 (R^2 \\:- r^2)dx = \\pi \\int^1_0 (x^2 + 4x + 4 \\:- (x^6 + 4x^3 + 4))dx = \\pi \\int^1_0 (-x^6 \\:- 4x^3 + x^2 + 4x) dx \\\\[5pt]<br \/>&amp; V = \\pi \\left[ -\\frac{1}{7}x^7 \\:- \\frac{4}{4}x^4 + \\frac{1}{3}x^3 + \\frac{4}{2}x^2 \\right] \\Bigg\\rvert^1_0 = \\pi \\left[ -\\frac{1}{7}x^7 \\:- x^4 + \\frac{1}{3}x^3 + 2x^2 \\right] \\Bigg\\rvert^1_0 \\\\[5pt]<br \/>&amp; V = \\pi \\left( \\left[ -\\frac{1}{7}(1)^7 \\:- (1)^4 + \\frac{1}{3}(1)^3 + 2(1)^2 \\right] \\:- \\left[ -\\frac{1}{7}(0)^7 \\:- (0)^4 + \\frac{1}{3}(0)^3 + 2(0)^2 \\right] \\right) \\\\[5pt]<br \/>&amp; V = \\pi \\left[ -\\frac{1}{7} \\:- 1 + \\frac{1}{3} + 2 \\right] \\\\[5pt]<br \/>&amp; \\color{green}{V =\\frac{25}{21}\\pi}<br \/>\\end{align*}<br \/>\\)<p>\u00a0<\/p><\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t\t<script type=\"application\/ld+json\">{\"@context\":\"https:\\\/\\\/schema.org\",\"@type\":\"FAQPage\",\"mainEntity\":[{\"@type\":\"Question\",\"name\":\"Problem 8: Solution\",\"acceptedAnswer\":{\"@type\":\"Answer\",\"text\":\"<h3>Compute the volume generated by rotating the region bounded by the graphs of \\\\(f(x)=x^3, g(x)=x \\\\le 0 \\\\le 1 \\\\):<\\\/h3><ul><li><h3>About x = 1<\\\/h3><\\\/li><\\\/ul><p><img class=\\\"wp-image-7108 size-medium aligncenter\\\" src=\\\"https:\\\/\\\/bisonmath.com\\\/wp-content\\\/uploads\\\/2022\\\/03\\\/Screenshot-43-300x300.png\\\" alt=\\\"\\\" width=\\\"300\\\" height=\\\"300\\\" \\\/><\\\/p><h4>Use the shell method because of the orientation of the differential line. Let \\\\(R = 1-x \\\\) and \\\\(x \\\\:- x^3\\\\). Updating these values in the integral yields:<\\\/h4>\\\\(<br \\\/>\\\\begin{align*}<br \\\/>&amp; V = \\\\pi \\\\int^1_0 (1-x)(x \\\\:- x^3)dx = \\\\pi \\\\int^1_0 (x^4 \\\\:- x^3 \\\\:- x^2 + x)dx \\\\\\\\[5pt]<br \\\/>&amp; V = \\\\pi \\\\left[ \\\\frac{1}{5}x^5 \\\\:- \\\\frac{1}{4}x^4 \\\\:- \\\\frac{1}{3}x^3 + \\\\frac{1}{2}x^2 \\\\right] \\\\Bigg\\\\rvert^1_0 \\\\\\\\[5pt]<br \\\/>&amp; V = \\\\pi \\\\left( \\\\left[ \\\\frac{1}{5}(1)^5 \\\\:- \\\\frac{1}{4}(1)^4 \\\\:- \\\\frac{1}{3}(1)^3 + \\\\frac{1}{2}(1)^2 \\\\right] \\\\:- \\\\left[ \\\\frac{1}{5}(0)^5 \\\\:- \\\\frac{1}{4}(0)^4 \\\\:- \\\\frac{1}{3}(0)^3 + \\\\frac{1}{2}(0)^2 \\\\right] \\\\right) \\\\\\\\[5pt]<br \\\/>&amp; V = \\\\pi \\\\left[ \\\\frac{1}{5} \\\\:- \\\\frac{1}{4} \\\\:- \\\\frac{1}{3} + \\\\frac{1}{2} \\\\right] \\\\\\\\[5pt]<br \\\/>&amp; \\\\color{green}{V = \\\\frac{7}{60}\\\\pi}<br \\\/>\\\\end{align*}<br \\\/>\\\\)<p>\\u00a0<\\\/p><ul><li><h3>About x = -2<\\\/h3><\\\/li><\\\/ul><p><img class=\\\"size-medium wp-image-7112 aligncenter\\\" src=\\\"https:\\\/\\\/bisonmath.com\\\/wp-content\\\/uploads\\\/2022\\\/03\\\/Screenshot-44-300x300.png\\\" alt=\\\"\\\" width=\\\"300\\\" height=\\\"300\\\" \\\/><\\\/p><h4>Once again, use the shell method. The problem is very similar to the last one. Let \\\\(R = x+2 \\\\) and \\\\(x \\\\:- x^3\\\\). Updating these values in the integral yields:<\\\/h4>\\\\(<br \\\/>\\\\begin{align*}<br \\\/>&amp; V = \\\\pi \\\\int^1_0 (x+2)(x \\\\:- x^3)dx = \\\\pi \\\\int^1_0 (-x^4 \\\\:- 2x^3 + x^2 + 2x)dx \\\\\\\\[5pt]<br \\\/>&amp; V = \\\\pi \\\\left[ -\\\\frac{1}{5}x^5 \\\\:- \\\\frac{2}{4}x^4 + \\\\frac{1}{3}x^3 + \\\\frac{2}{2}x^2 \\\\right] \\\\Bigg\\\\rvert^1_0\\u00a0 = \\\\pi \\\\left[ -\\\\frac{1}{5}x^5 \\\\:- \\\\frac{1}{2}x^4 + \\\\frac{1}{3}x^3 + x^2 \\\\right] \\\\Bigg\\\\rvert^1_0 \\\\\\\\[5pt]<br \\\/>&amp; V = \\\\pi \\\\left( \\\\left[ -\\\\frac{1}{5}(1)^5 \\\\:- \\\\frac{1}{2}(1)^4 + \\\\frac{1}{3}(1)^3 + (1)^2 \\\\right] \\\\:- \\\\left[ -\\\\frac{1}{5}(0)^5 \\\\:- \\\\frac{1}{2}(0)^4 + \\\\frac{1}{3}(0)^3 + (0)^2 \\\\right] \\\\right) \\\\\\\\[5pt]<br \\\/>&amp; V = \\\\pi \\\\left[ -\\\\frac{1}{5} \\\\:- \\\\frac{1}{2} + \\\\frac{1}{3} + 1 \\\\right] \\\\\\\\[5pt]<br \\\/>&amp; \\\\color{green}{V = \\\\frac{19}{30}\\\\pi}<br \\\/>\\\\end{align*}<br \\\/>\\\\)<p>\\u00a0<\\\/p><ul><li><h3>About y = 2<\\\/h3><\\\/li><\\\/ul><p><img class=\\\"size-medium wp-image-7132 aligncenter\\\" src=\\\"https:\\\/\\\/bisonmath.com\\\/wp-content\\\/uploads\\\/2022\\\/03\\\/Screenshot-45-300x300.png\\\" alt=\\\"\\\" width=\\\"300\\\" height=\\\"300\\\" \\\/><\\\/p><h4>Given the orientation of the differential line with respect to the axis of rotation, use the washer method:<\\\/h4>\\\\(<br \\\/>\\\\begin{align*}<br \\\/>&amp; R = 2 \\\\:- x^3; \\\\Rightarrow R^2 = (2 \\\\:- x^3)^2 = 4 \\\\:- 4x^3 + x^6 \\\\\\\\[5pt]<br \\\/>&amp; r = 2 \\\\:- x; \\\\Rightarrow r^2 = (2 \\\\:- x)^2 = 4 \\\\:- 4x + x^2 \\\\\\\\[10pt]<br \\\/>&amp; V = \\\\pi \\\\int^1_0 (R^2 \\\\:- r^2)dx = \\\\pi \\\\int^1_0 (4 \\\\:- 4x^3 + x^6 \\\\:- (4 \\\\:- 4x + x^2)) dx = \\\\pi \\\\int^1_0 (x^6 \\\\:- 4x^3 \\\\:- x^2 + 4x)dx \\\\\\\\[5pt]<br \\\/>&amp; V = \\\\pi \\\\left[ \\\\frac{1}{7}x^7 \\\\:- \\\\frac{4}{4}x^4 \\\\:- \\\\frac{1}{3}x^3 + \\\\frac{4}{2}x^2 \\\\right] \\\\Bigg\\\\rvert^1_0 = \\\\pi \\\\left[ \\\\frac{1}{7}x^7 \\\\:- x^4 \\\\:- \\\\frac{1}{3}x^3 + 2x^2 \\\\right] \\\\Bigg\\\\rvert^1_0 \\\\\\\\[5pt]<br \\\/>&amp; V = \\\\pi \\\\left( \\\\left[ \\\\frac{1}{7}(1)^7 \\\\:- (1)^4 \\\\:- \\\\frac{1}{3}(1)^3 + 2(1)^2 \\\\right] \\\\:- \\\\left[ \\\\frac{1}{7}(0)^7 \\\\:- (0)^4 \\\\:- \\\\frac{1}{3}(0)^3 + 2(0)^2 \\\\right] \\\\right) \\\\\\\\[5pt]<br \\\/>&amp; V = \\\\pi \\\\left[ \\\\frac{1}{7} \\\\:- 1 \\\\:- \\\\frac{1}{3} + 2 \\\\right] \\\\\\\\[5pt]<br \\\/>&amp; \\\\color{green}{V = \\\\frac{17}{21}\\\\pi}<br \\\/>\\\\end{align*}<br \\\/>\\\\)<p>\\u00a0<\\\/p><ul><li><h3>About y = -2<\\\/h3><\\\/li><\\\/ul><p><img class=\\\"size-medium wp-image-7171 aligncenter\\\" src=\\\"https:\\\/\\\/bisonmath.com\\\/wp-content\\\/uploads\\\/2022\\\/03\\\/Screenshot-46-300x300.png\\\" alt=\\\"\\\" width=\\\"300\\\" height=\\\"300\\\" \\\/><\\\/p><h4>Once again, apply the washer method because of the orientation of the differential line:<\\\/h4>\\\\(<br \\\/>\\\\begin{align*}<br \\\/>&amp; R = x + 2; \\\\Rightarrow R^2 = (x + 2)^2 = x^2 + 4x + 4 \\\\\\\\[5pt]<br \\\/>&amp; r = x^3 + 2; \\\\Rightarrow r^2 = (x^3 + 2)^2 = x^6 + 4x^3 + 4 \\\\\\\\[10pt]<br \\\/>&amp; V = \\\\pi \\\\int^1_0 (R^2 \\\\:- r^2)dx = \\\\pi \\\\int^1_0 (x^2 + 4x + 4 \\\\:- (x^6 + 4x^3 + 4))dx = \\\\pi \\\\int^1_0 (-x^6 \\\\:- 4x^3 + x^2 + 4x) dx \\\\\\\\[5pt]<br \\\/>&amp; V = \\\\pi \\\\left[ -\\\\frac{1}{7}x^7 \\\\:- \\\\frac{4}{4}x^4 + \\\\frac{1}{3}x^3 + \\\\frac{4}{2}x^2 \\\\right] \\\\Bigg\\\\rvert^1_0 = \\\\pi \\\\left[ -\\\\frac{1}{7}x^7 \\\\:- x^4 + \\\\frac{1}{3}x^3 + 2x^2 \\\\right] \\\\Bigg\\\\rvert^1_0 \\\\\\\\[5pt]<br \\\/>&amp; V = \\\\pi \\\\left( \\\\left[ -\\\\frac{1}{7}(1)^7 \\\\:- (1)^4 + \\\\frac{1}{3}(1)^3 + 2(1)^2 \\\\right] \\\\:- \\\\left[ -\\\\frac{1}{7}(0)^7 \\\\:- (0)^4 + \\\\frac{1}{3}(0)^3 + 2(0)^2 \\\\right] \\\\right) \\\\\\\\[5pt]<br \\\/>&amp; V = \\\\pi \\\\left[ -\\\\frac{1}{7} \\\\:- 1 + \\\\frac{1}{3} + 2 \\\\right] \\\\\\\\[5pt]<br \\\/>&amp; \\\\color{green}{V =\\\\frac{25}{21}\\\\pi}<br \\\/>\\\\end{align*}<br \\\/>\\\\)<p>\\u00a0<\\\/p>\"}}]}<\/script>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-59174df3 elementor-widget elementor-widget-toggle\" data-id=\"59174df3\" data-element_type=\"widget\" data-widget_type=\"toggle.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<div class=\"elementor-toggle\">\n\t\t\t\t\t\t\t<div class=\"elementor-toggle-item\">\n\t\t\t\t\t<h2 id=\"elementor-tab-title-1491\" class=\"elementor-tab-title\" data-tab=\"1\" role=\"button\" aria-controls=\"elementor-tab-content-1491\" aria-expanded=\"false\">\n\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon elementor-toggle-icon-left\" aria-hidden=\"true\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon-closed\"><i class=\"fas fa-toggle-off\"><\/i><\/span>\n\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon-opened\"><i class=\"elementor-toggle-icon-opened fas fa-toggle-on\"><\/i><\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t\t<\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t<a class=\"elementor-toggle-title\" tabindex=\"0\">Problem 9: Solution<\/a>\n\t\t\t\t\t<\/h2>\n\n\t\t\t\t\t<div id=\"elementor-tab-content-1491\" class=\"elementor-tab-content elementor-clearfix\" data-tab=\"1\" role=\"region\" aria-labelledby=\"elementor-tab-title-1491\"><h3>Compute the volume generated by rotating the region bounded by the graphs of the given equations:<\/h3><ul><li><h3>(a) About x=-1<\/h3><\/li><\/ul><h4>First, sketch all of the given functions and identify a common area among them:<\/h4><p><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-7197 size-medium aligncenter\" src=\"https:\/\/bisonmath.com\/wp-content\/uploads\/2022\/03\/Screenshot-47-300x300.png\" alt=\"\" width=\"300\" height=\"300\" \/><\/p><h4>Finally, choose the proper method of volumetric intergation. Notice how the differencial line is parallel to the axis of rotation. Because of this, use the shell method: Let \\(R = x + 1\\) and \\(H = (x \\:- 2)^3 \\:- (x \\:- 2)= x^3 \\:- 6x^2 + 11x \\:- 6 \\)<\/h4>\\(<br \/>\\begin{align*}<br \/>&amp; V = 2\\pi \\int^2_1 [(x+1)(x^3 \\:- 6x^2 + 11x \\:- 6)]dx = 2\\pi \\int^1_0 [x^4 \\:- 5x^3 + 5x^2 + 5x \\:- 6]dx \\\\[5pt]<br \/>&amp; V = 2\\pi \\left[ \\frac{1}{5}x^5 \\:- \\frac{5}{4}x^4 + \\frac{5}{3}x^3 + \\frac{5}{2}x^2 \\:- 6x \\right] \\Bigg\\rvert^2_1 \\\\[5pt]<br \/>&amp; V = 2\\pi \\left[ \\left( \\frac{1}{5}(2)^5 \\:- \\frac{5}{4}(2)^4 + \\frac{5}{3}(2)^3 + \\frac{5}{2}(2)^2 \\:- 6(2) \\right) \\:- \\left( \\frac{1}{5}(1)^5 \\:- \\frac{5}{4}(1)^4 + \\frac{5}{3}(1)^3 + \\frac{5}{2}(1)^2 \\:- 6(1)\\right) \\right] \\\\[5pt]<br \/>&amp; V = 2\\pi \\left[ \\left( \\frac{32}{5} \\:- 20 + \\frac{40}{3} + 10 \\:- 12 \\right) \\:- \\left( \\frac{1}{5} \\:- \\frac{5}{4} + \\frac{5}{3} + \\frac{5}{2} \\:- 6 \\right) \\right] = 2\\pi \\left[-\\frac{34}{15} -\\left(-\\frac{173}{60}\\right)\\right] \\\\[5pt]<br \/>&amp; \\color{green}{V = \\frac{37}{30}\\pi}<br \/>\\end{align*}<br \/>\\)<p>\u00a0<\/p><ul><li><h3>(b) About x=2<\/h3><\/li><\/ul><h4>First, sketch all of the given functions and identify a common area among them:<\/h4><h4><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-7226 size-medium aligncenter\" src=\"https:\/\/bisonmath.com\/wp-content\/uploads\/2022\/03\/Screenshot-48-300x300.png\" alt=\"\" width=\"300\" height=\"300\" \/><\/h4><h4>The problem remains mostly unchanged, the only modification being the radius \\(R = 2 \\:- x\\). Updating the previous integral with this change yields:<\/h4>\\(<br \/>\\begin{align*}<br \/>&amp; V = 2\\pi \\int^2_1 (2 \\:- x)(x^3 \\:- 6x^2 + 11x \\:- 6) dx = 2\\pi \\int^1_0 (-x^4 + 8x^3 \\:- 23x^2 + 28x \\:- 12)dx \\\\[5pt]<br \/>&amp; V = 2\\pi \\left[ -\\frac{1}{5}x^5 + \\frac{8}{4}x^4 \\:- \\frac{23}{3}x^3 +\\frac{28}{2}x^2 \\:- 12x \\right] \\Bigg\\rvert^2_1 = 2\\pi \\left[ -\\frac{1}{5}x^5 + 2x^4 \\:- \\frac{23}{3}x^3 + 14x^2 \\:- 12x \\right] \\Bigg\\rvert^2_1 \\\\[5pt]<br \/>&amp; V = 2\\pi \\left( \\left[ -\\frac{1}{5}(2)^5 + 2(2)^4 \\:- \\frac{23}{3}(2)^3 + 14(2)^2 \\:- 12(2) \\right] \\:- \\left[ -\\frac{1}{5}(1)^5 + 2(1)^4 \\:- \\frac{23}{3}(1)^3 + 14(1)^2 \\:- 12(1) \\right] \\right) \\\\[5pt]<br \/>&amp; V = 2\\pi \\left[ \\left(-\\frac{32}{5} + 32 \\:- \\frac{184}{3} + 56 \\:- 24 \\right) \\:- \\left(-\\frac{1}{5} + 2 \\:- \\frac{23}{3} + 14 \\:- 12\\right) \\right] = 2\\pi\\left[-\\frac{56}{15}-\\left(-\\frac{58}{15}\\right)\\right] \\\\[5pt]<br \/>&amp; \\color{green}{V = \\frac{4}{15}\\pi}<br \/>\\end{align*}<br \/>\\)<p>\u00a0<\/p><ul><li><h3>(c) About y =-1<\/h3><\/li><\/ul><h4>First, sketch all of the given functions and identify a common area among them:<\/h4><p><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-7245 size-medium aligncenter\" src=\"https:\/\/bisonmath.com\/wp-content\/uploads\/2022\/03\/Screenshot-49-300x300.png\" alt=\"\" width=\"300\" height=\"300\" \/><\/p><h4>Now, as we can&#8217;t change the orientation of the differential line, change the method to washer (because of the perperndicular position between the differential and the rotation axis):<\/h4>\\(<br \/>\\begin{align*}<br \/>&amp; R = (x \\:- 2)^3 + 1; \\Rightarrow R^2 = ((x \\:- 2)^3 + 1)^2 = x^6 \\:- 12x^5 + 60x^4 \\:-158x^3 + 228x^2 \\:- 168x + 49 \\\\[5pt]<br \/>&amp; r = (x \\:- 2) + 1; \\Rightarrow r = (x \\:- 1)^2 = x^2 \\:- 2x + 1 \\\\[10pt]<br \/>&amp; R^2 \\:- r^2 = x^6 \\:- 12x^5 + 60x^4 \\:- 158x^3 + 228x^2 \\:- 168x + 49 \\:- ( x^2 \\:- 2x + 1) \\\\[5pt]<br \/>&amp; = x^6 \\:- 12x^5 + 60x^4 \\:-158x^3 + 227x^2 \\:- 166x+ 48 \\\\\\\\[5pt]<br \/>&amp; V = \\pi \\int^2_1 \\left( x^6 \\:- 12x^5 + 60x^4 \\:-158x^3 + 227x^2 \\:- 166x+ 48 \\right)dx \\\\[5pt]<br \/>&amp; V = \\pi\\left[ \\frac{1}{7}x^7 \\:- \\frac{12}{6}x^6 + \\frac{60}{5}x^5 \\:- \\frac{158}{4}x^4 + \\frac{227}{3}x^3 \\:- \\frac{166}{2}x^2 + 48x \\right] \\Bigg\\rvert^2_1 \\\\[5pt]<br \/>&amp; V = \\pi\\left[ \\frac{1}{7}x^7 \\:- 2x^6 + 12x^5 \\:- \\frac{79}{2}x^4 + \\frac{227}{3}x^3 \\:- 83x^2 + 48x \\right] \\Bigg\\rvert^2_1 \\\\[5pt]<br \/>&amp; V = \\pi\\left( \\left[ \\frac{1}{7}(2)^7 \\:- 2(2)^6 + 12(2)^5 \\:- \\frac{79}{2}(2)^4 + \\frac{227}{3}(2)^3 \\:- 83(2)^2 + 48(2) \\right] \\:- \\left[ \\frac{1}{7}(1)^7 \\:- 2(1)^6 + 12(1)^5 \\:- \\frac{79}{2}(1)^4 + \\frac{227}{3}(1)^3 \\:- 83(1)^2 + 48(1) \\right] \\right) \\\\[5pt]<br \/>&amp; V = \\pi \\left[ \\left( \\frac{128}{7} \\:- 128 + 384 \\:- 632 + \\frac{1816}{3} \\:- 332 + 96 \\right) \\:- \\left( \\frac{1}{7} \\:- 2 + 12 \\:- \\frac{79}{2} + \\frac{227}{3} \\:- 83 + 48 \\right) \\right] = \\pi \\left[ \\left(\\frac{244}{21}\\right) \\:- \\left(\\frac{475}{12} \\right) \\right] \\\\[5pt]<br \/>&amp; \\color{green}{V = \\frac{13}{42}\\pi}<br \/>\\end{align*}<br \/>\\)<p>\u00a0<\/p><ul><li><h3>(d) About y = 1<\/h3><\/li><\/ul><h4>First, sketch all of the given functions and identify a common area among them:<\/h4><p><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-7282 size-medium aligncenter\" src=\"https:\/\/bisonmath.com\/wp-content\/uploads\/2022\/03\/Screenshot-50-300x300.png\" alt=\"\" width=\"300\" height=\"300\" \/><\/p><h4>The problem remains mostly unchanged (we are still using the washer method), so the only modification being both radiuses:<\/h4>\\(<br \/>\\begin{align*}<br \/>&amp; R = 1 \\:- (x \\:- 2); \\Rightarrow r = (3 \\:- x)^2 = 9 \\:- 6x + x^2 \\\\[10pt]<br \/>&amp; r = 1 \\:- (x \\:- 2)^3; \\Rightarrow R^2 = (1 \\:- (x \\:- 2)^3)^2 = x^6 \\:- 12x^5 + 60x^4 \\:- 162x^3 + 252x^2 \\:- 216x + 81 \\\\[5pt]<br \/>&amp; R^2 \\:- r^2 = ( 9 \\:- 6x + x^2) \\:- (x^6 \\:- 12x^5 + 60x^4 \\:- 162x^3 + 252x^2 \\:- 216x + 81) \\\\[5pt]<br \/>&amp; = -x^6 + 12x^5 \\:- 60x^4 + 162x^3 \\:- 251x^2 + 210x \\:- 72 \\\\\\\\[5pt]<br \/>&amp; V = \\pi \\int^2_1 \\left( -x^6 + 12x^5 \\:- 60x^4 + 162x^3 \\:- 251x^2 + 210x \\:- 72 \\right)dx \\\\[5pt]<br \/>&amp; V = \\pi\\left[ -\\frac{1}{7}x^7 + \\frac{12}{6}x^6 \\:- \\frac{60}{5}x^5 + \\frac{162}{4}x^4 \\:- \\frac{251}{3}x^3 + \\frac{210}{2}x^2 \\:- 72x \\right] \\Bigg\\rvert^2_1 \\\\[5pt]<br \/>&amp; V = \\pi\\left[ -\\frac{1}{7}x^7 + 2x^6 \\:- 12x^5 + \\frac{81}{2}x^4 \\:- \\frac{251}{3}x^3 + 105x^2 \\:- 72x \\right] \\Bigg\\rvert^2_1 \\\\[5pt]<br \/>&amp; V = \\pi\\left( \\left[ -\\frac{1}{7}(2)^7 + 2(2)^6 \\:- 12(2)^5 + \\frac{81}{2}(2)^4 \\:- \\frac{251}{3}(2)^3 + 105(2)^2 \\:- 72(2) \\right] \\:- \\left[ -\\frac{1}{7}(1)^7 + 2(1)^6 \\:- 12(1)^5 + \\frac{81}{2}(1)^4 \\:- \\frac{251}{3}(1)^3 + 105(1)^2 \\:- 72(1) \\right] \\right) \\\\[5pt]<br \/>&amp; V = \\pi \\left( \\left[ -\\frac{128}{7} + 128 \\:- 384 + 648 \\:- \\frac{2008}{3} + 420 \\:- 144 \\right] \\:- \\left[ -\\frac{1}{7} + 2 \\:- 12 + \\frac{81}{2} \\:- \\frac{251}{3} + 105 \\:- 72 \\right] \\right) \\\\[5pt]<br \/>&amp; V = \\pi\\left( \\left[ -\\frac{412}{21} \\right] \\:- \\left[ -\\frac{853}{42} \\right] \\right) \\\\[5pt]<br \/>&amp; \\color{green}{V = \\frac{29}{42}\\pi}<br \/>\\end{align*}<br \/>\\)<p>\u00a0<\/p><\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t\t<script type=\"application\/ld+json\">{\"@context\":\"https:\\\/\\\/schema.org\",\"@type\":\"FAQPage\",\"mainEntity\":[{\"@type\":\"Question\",\"name\":\"Problem 9: Solution\",\"acceptedAnswer\":{\"@type\":\"Answer\",\"text\":\"<h3>Compute the volume generated by rotating the region bounded by the graphs of the given equations:<\\\/h3><ul><li><h3>(a) About x=-1<\\\/h3><\\\/li><\\\/ul><h4>First, sketch all of the given functions and identify a common area among them:<\\\/h4><p><img class=\\\"wp-image-7197 size-medium aligncenter\\\" src=\\\"https:\\\/\\\/bisonmath.com\\\/wp-content\\\/uploads\\\/2022\\\/03\\\/Screenshot-47-300x300.png\\\" alt=\\\"\\\" width=\\\"300\\\" height=\\\"300\\\" \\\/><\\\/p><h4>Finally, choose the proper method of volumetric intergation. Notice how the differencial line is parallel to the axis of rotation. Because of this, use the shell method: Let \\\\(R = x + 1\\\\) and \\\\(H = (x \\\\:- 2)^3 \\\\:- (x \\\\:- 2)= x^3 \\\\:- 6x^2 + 11x \\\\:- 6 \\\\)<\\\/h4>\\\\(<br \\\/>\\\\begin{align*}<br \\\/>&amp; V = 2\\\\pi \\\\int^2_1 [(x+1)(x^3 \\\\:- 6x^2 + 11x \\\\:- 6)]dx = 2\\\\pi \\\\int^1_0 [x^4 \\\\:- 5x^3 + 5x^2 + 5x \\\\:- 6]dx \\\\\\\\[5pt]<br \\\/>&amp; V = 2\\\\pi \\\\left[ \\\\frac{1}{5}x^5 \\\\:- \\\\frac{5}{4}x^4 + \\\\frac{5}{3}x^3 + \\\\frac{5}{2}x^2 \\\\:- 6x \\\\right] \\\\Bigg\\\\rvert^2_1 \\\\\\\\[5pt]<br \\\/>&amp; V = 2\\\\pi \\\\left[ \\\\left( \\\\frac{1}{5}(2)^5 \\\\:- \\\\frac{5}{4}(2)^4 + \\\\frac{5}{3}(2)^3 + \\\\frac{5}{2}(2)^2 \\\\:- 6(2) \\\\right) \\\\:- \\\\left( \\\\frac{1}{5}(1)^5 \\\\:- \\\\frac{5}{4}(1)^4 + \\\\frac{5}{3}(1)^3 + \\\\frac{5}{2}(1)^2 \\\\:- 6(1)\\\\right) \\\\right] \\\\\\\\[5pt]<br \\\/>&amp; V = 2\\\\pi \\\\left[ \\\\left( \\\\frac{32}{5} \\\\:- 20 + \\\\frac{40}{3} + 10 \\\\:- 12 \\\\right) \\\\:- \\\\left( \\\\frac{1}{5} \\\\:- \\\\frac{5}{4} + \\\\frac{5}{3} + \\\\frac{5}{2} \\\\:- 6 \\\\right) \\\\right] = 2\\\\pi \\\\left[-\\\\frac{34}{15} -\\\\left(-\\\\frac{173}{60}\\\\right)\\\\right] \\\\\\\\[5pt]<br \\\/>&amp; \\\\color{green}{V = \\\\frac{37}{30}\\\\pi}<br \\\/>\\\\end{align*}<br \\\/>\\\\)<p>\\u00a0<\\\/p><ul><li><h3>(b) About x=2<\\\/h3><\\\/li><\\\/ul><h4>First, sketch all of the given functions and identify a common area among them:<\\\/h4><h4><img class=\\\"wp-image-7226 size-medium aligncenter\\\" src=\\\"https:\\\/\\\/bisonmath.com\\\/wp-content\\\/uploads\\\/2022\\\/03\\\/Screenshot-48-300x300.png\\\" alt=\\\"\\\" width=\\\"300\\\" height=\\\"300\\\" \\\/><\\\/h4><h4>The problem remains mostly unchanged, the only modification being the radius \\\\(R = 2 \\\\:- x\\\\). Updating the previous integral with this change yields:<\\\/h4>\\\\(<br \\\/>\\\\begin{align*}<br \\\/>&amp; V = 2\\\\pi \\\\int^2_1 (2 \\\\:- x)(x^3 \\\\:- 6x^2 + 11x \\\\:- 6) dx = 2\\\\pi \\\\int^1_0 (-x^4 + 8x^3 \\\\:- 23x^2 + 28x \\\\:- 12)dx \\\\\\\\[5pt]<br \\\/>&amp; V = 2\\\\pi \\\\left[ -\\\\frac{1}{5}x^5 + \\\\frac{8}{4}x^4 \\\\:- \\\\frac{23}{3}x^3 +\\\\frac{28}{2}x^2 \\\\:- 12x \\\\right] \\\\Bigg\\\\rvert^2_1 = 2\\\\pi \\\\left[ -\\\\frac{1}{5}x^5 + 2x^4 \\\\:- \\\\frac{23}{3}x^3 + 14x^2 \\\\:- 12x \\\\right] \\\\Bigg\\\\rvert^2_1 \\\\\\\\[5pt]<br \\\/>&amp; V = 2\\\\pi \\\\left( \\\\left[ -\\\\frac{1}{5}(2)^5 + 2(2)^4 \\\\:- \\\\frac{23}{3}(2)^3 + 14(2)^2 \\\\:- 12(2) \\\\right] \\\\:- \\\\left[ -\\\\frac{1}{5}(1)^5 + 2(1)^4 \\\\:- \\\\frac{23}{3}(1)^3 + 14(1)^2 \\\\:- 12(1) \\\\right] \\\\right) \\\\\\\\[5pt]<br \\\/>&amp; V = 2\\\\pi \\\\left[ \\\\left(-\\\\frac{32}{5} + 32 \\\\:- \\\\frac{184}{3} + 56 \\\\:- 24 \\\\right) \\\\:- \\\\left(-\\\\frac{1}{5} + 2 \\\\:- \\\\frac{23}{3} + 14 \\\\:- 12\\\\right) \\\\right] = 2\\\\pi\\\\left[-\\\\frac{56}{15}-\\\\left(-\\\\frac{58}{15}\\\\right)\\\\right] \\\\\\\\[5pt]<br \\\/>&amp; \\\\color{green}{V = \\\\frac{4}{15}\\\\pi}<br \\\/>\\\\end{align*}<br \\\/>\\\\)<p>\\u00a0<\\\/p><ul><li><h3>(c) About y =-1<\\\/h3><\\\/li><\\\/ul><h4>First, sketch all of the given functions and identify a common area among them:<\\\/h4><p><img class=\\\"wp-image-7245 size-medium aligncenter\\\" src=\\\"https:\\\/\\\/bisonmath.com\\\/wp-content\\\/uploads\\\/2022\\\/03\\\/Screenshot-49-300x300.png\\\" alt=\\\"\\\" width=\\\"300\\\" height=\\\"300\\\" \\\/><\\\/p><h4>Now, as we can&#8217;t change the orientation of the differential line, change the method to washer (because of the perperndicular position between the differential and the rotation axis):<\\\/h4>\\\\(<br \\\/>\\\\begin{align*}<br \\\/>&amp; R = (x \\\\:- 2)^3 + 1; \\\\Rightarrow R^2 = ((x \\\\:- 2)^3 + 1)^2 = x^6 \\\\:- 12x^5 + 60x^4 \\\\:-158x^3 + 228x^2 \\\\:- 168x + 49 \\\\\\\\[5pt]<br \\\/>&amp; r = (x \\\\:- 2) + 1; \\\\Rightarrow r = (x \\\\:- 1)^2 = x^2 \\\\:- 2x + 1 \\\\\\\\[10pt]<br \\\/>&amp; R^2 \\\\:- r^2 = x^6 \\\\:- 12x^5 + 60x^4 \\\\:- 158x^3 + 228x^2 \\\\:- 168x + 49 \\\\:- ( x^2 \\\\:- 2x + 1) \\\\\\\\[5pt]<br \\\/>&amp; = x^6 \\\\:- 12x^5 + 60x^4 \\\\:-158x^3 + 227x^2 \\\\:- 166x+ 48 \\\\\\\\\\\\\\\\[5pt]<br \\\/>&amp; V = \\\\pi \\\\int^2_1 \\\\left( x^6 \\\\:- 12x^5 + 60x^4 \\\\:-158x^3 + 227x^2 \\\\:- 166x+ 48 \\\\right)dx \\\\\\\\[5pt]<br \\\/>&amp; V = \\\\pi\\\\left[ \\\\frac{1}{7}x^7 \\\\:- \\\\frac{12}{6}x^6 + \\\\frac{60}{5}x^5 \\\\:- \\\\frac{158}{4}x^4 + \\\\frac{227}{3}x^3 \\\\:- \\\\frac{166}{2}x^2 + 48x \\\\right] \\\\Bigg\\\\rvert^2_1 \\\\\\\\[5pt]<br \\\/>&amp; V = \\\\pi\\\\left[ \\\\frac{1}{7}x^7 \\\\:- 2x^6 + 12x^5 \\\\:- \\\\frac{79}{2}x^4 + \\\\frac{227}{3}x^3 \\\\:- 83x^2 + 48x \\\\right] \\\\Bigg\\\\rvert^2_1 \\\\\\\\[5pt]<br \\\/>&amp; V = \\\\pi\\\\left( \\\\left[ \\\\frac{1}{7}(2)^7 \\\\:- 2(2)^6 + 12(2)^5 \\\\:- \\\\frac{79}{2}(2)^4 + \\\\frac{227}{3}(2)^3 \\\\:- 83(2)^2 + 48(2) \\\\right] \\\\:- \\\\left[ \\\\frac{1}{7}(1)^7 \\\\:- 2(1)^6 + 12(1)^5 \\\\:- \\\\frac{79}{2}(1)^4 + \\\\frac{227}{3}(1)^3 \\\\:- 83(1)^2 + 48(1) \\\\right] \\\\right) \\\\\\\\[5pt]<br \\\/>&amp; V = \\\\pi \\\\left[ \\\\left( \\\\frac{128}{7} \\\\:- 128 + 384 \\\\:- 632 + \\\\frac{1816}{3} \\\\:- 332 + 96 \\\\right) \\\\:- \\\\left( \\\\frac{1}{7} \\\\:- 2 + 12 \\\\:- \\\\frac{79}{2} + \\\\frac{227}{3} \\\\:- 83 + 48 \\\\right) \\\\right] = \\\\pi \\\\left[ \\\\left(\\\\frac{244}{21}\\\\right) \\\\:- \\\\left(\\\\frac{475}{12} \\\\right) \\\\right] \\\\\\\\[5pt]<br \\\/>&amp; \\\\color{green}{V = \\\\frac{13}{42}\\\\pi}<br \\\/>\\\\end{align*}<br \\\/>\\\\)<p>\\u00a0<\\\/p><ul><li><h3>(d) About y = 1<\\\/h3><\\\/li><\\\/ul><h4>First, sketch all of the given functions and identify a common area among them:<\\\/h4><p><img class=\\\"wp-image-7282 size-medium aligncenter\\\" src=\\\"https:\\\/\\\/bisonmath.com\\\/wp-content\\\/uploads\\\/2022\\\/03\\\/Screenshot-50-300x300.png\\\" alt=\\\"\\\" width=\\\"300\\\" height=\\\"300\\\" \\\/><\\\/p><h4>The problem remains mostly unchanged (we are still using the washer method), so the only modification being both radiuses:<\\\/h4>\\\\(<br \\\/>\\\\begin{align*}<br \\\/>&amp; R = 1 \\\\:- (x \\\\:- 2); \\\\Rightarrow r = (3 \\\\:- x)^2 = 9 \\\\:- 6x + x^2 \\\\\\\\[10pt]<br \\\/>&amp; r = 1 \\\\:- (x \\\\:- 2)^3; \\\\Rightarrow R^2 = (1 \\\\:- (x \\\\:- 2)^3)^2 = x^6 \\\\:- 12x^5 + 60x^4 \\\\:- 162x^3 + 252x^2 \\\\:- 216x + 81 \\\\\\\\[5pt]<br \\\/>&amp; R^2 \\\\:- r^2 = ( 9 \\\\:- 6x + x^2) \\\\:- (x^6 \\\\:- 12x^5 + 60x^4 \\\\:- 162x^3 + 252x^2 \\\\:- 216x + 81) \\\\\\\\[5pt]<br \\\/>&amp; = -x^6 + 12x^5 \\\\:- 60x^4 + 162x^3 \\\\:- 251x^2 + 210x \\\\:- 72 \\\\\\\\\\\\\\\\[5pt]<br \\\/>&amp; V = \\\\pi \\\\int^2_1 \\\\left( -x^6 + 12x^5 \\\\:- 60x^4 + 162x^3 \\\\:- 251x^2 + 210x \\\\:- 72 \\\\right)dx \\\\\\\\[5pt]<br \\\/>&amp; V = \\\\pi\\\\left[ -\\\\frac{1}{7}x^7 + \\\\frac{12}{6}x^6 \\\\:- \\\\frac{60}{5}x^5 + \\\\frac{162}{4}x^4 \\\\:- \\\\frac{251}{3}x^3 + \\\\frac{210}{2}x^2 \\\\:- 72x \\\\right] \\\\Bigg\\\\rvert^2_1 \\\\\\\\[5pt]<br \\\/>&amp; V = \\\\pi\\\\left[ -\\\\frac{1}{7}x^7 + 2x^6 \\\\:- 12x^5 + \\\\frac{81}{2}x^4 \\\\:- \\\\frac{251}{3}x^3 + 105x^2 \\\\:- 72x \\\\right] \\\\Bigg\\\\rvert^2_1 \\\\\\\\[5pt]<br \\\/>&amp; V = \\\\pi\\\\left( \\\\left[ -\\\\frac{1}{7}(2)^7 + 2(2)^6 \\\\:- 12(2)^5 + \\\\frac{81}{2}(2)^4 \\\\:- \\\\frac{251}{3}(2)^3 + 105(2)^2 \\\\:- 72(2) \\\\right] \\\\:- \\\\left[ -\\\\frac{1}{7}(1)^7 + 2(1)^6 \\\\:- 12(1)^5 + \\\\frac{81}{2}(1)^4 \\\\:- \\\\frac{251}{3}(1)^3 + 105(1)^2 \\\\:- 72(1) \\\\right] \\\\right) \\\\\\\\[5pt]<br \\\/>&amp; V = \\\\pi \\\\left( \\\\left[ -\\\\frac{128}{7} + 128 \\\\:- 384 + 648 \\\\:- \\\\frac{2008}{3} + 420 \\\\:- 144 \\\\right] \\\\:- \\\\left[ -\\\\frac{1}{7} + 2 \\\\:- 12 + \\\\frac{81}{2} \\\\:- \\\\frac{251}{3} + 105 \\\\:- 72 \\\\right] \\\\right) \\\\\\\\[5pt]<br \\\/>&amp; V = \\\\pi\\\\left( \\\\left[ -\\\\frac{412}{21} \\\\right] \\\\:- \\\\left[ -\\\\frac{853}{42} \\\\right] \\\\right) \\\\\\\\[5pt]<br \\\/>&amp; \\\\color{green}{V = \\\\frac{29}{42}\\\\pi}<br \\\/>\\\\end{align*}<br \\\/>\\\\)<p>\\u00a0<\\\/p>\"}}]}<\/script>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-363a85c9 elementor-widget elementor-widget-toggle\" data-id=\"363a85c9\" data-element_type=\"widget\" data-widget_type=\"toggle.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<div class=\"elementor-toggle\">\n\t\t\t\t\t\t\t<div class=\"elementor-toggle-item\">\n\t\t\t\t\t<h2 id=\"elementor-tab-title-9091\" class=\"elementor-tab-title\" data-tab=\"1\" role=\"button\" aria-controls=\"elementor-tab-content-9091\" aria-expanded=\"false\">\n\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon elementor-toggle-icon-left\" aria-hidden=\"true\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon-closed\"><i class=\"fas fa-toggle-off\"><\/i><\/span>\n\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon-opened\"><i class=\"elementor-toggle-icon-opened fas fa-toggle-on\"><\/i><\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t\t<\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t<a class=\"elementor-toggle-title\" tabindex=\"0\">Problem 10: Solution<\/a>\n\t\t\t\t\t<\/h2>\n\n\t\t\t\t\t<div id=\"elementor-tab-content-9091\" class=\"elementor-tab-content elementor-clearfix\" data-tab=\"1\" role=\"region\" aria-labelledby=\"elementor-tab-title-9091\"><h3>Describe the solid whose volume is being represented by the given integral:<\/h3>\\(<br \/>\\begin{align*}<br \/>2\\pi \\int^1_0 (2x-x^2) dx = 2\\pi \\int^1_0 x(2-x) dx<br \/>\\end{align*}<br \/>\\)<p>\u00a0<\/p><h4>First, we start with the original given integral. Then, we factor the term x out of the integrand. Since we have \\(2\\pi\\) at the beginning of the integral, we can safely assume that this is a shell method integral. The radius would be the factored \\(x\\), the height would be \\(2-x\\), and the limits of integration would be \\(0 \\le x \\le 1\\). The graph is as follows:<\/h4><p><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-6695 aligncenter\" src=\"https:\/\/bisonmath.com\/wp-content\/uploads\/2022\/03\/Screenshot-29-300x300.png\" alt=\"\" width=\"300\" height=\"300\" \/><\/p><\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t\t<script type=\"application\/ld+json\">{\"@context\":\"https:\\\/\\\/schema.org\",\"@type\":\"FAQPage\",\"mainEntity\":[{\"@type\":\"Question\",\"name\":\"Problem 10: Solution\",\"acceptedAnswer\":{\"@type\":\"Answer\",\"text\":\"<h3>Describe the solid whose volume is being represented by the given integral:<\\\/h3>\\\\(<br \\\/>\\\\begin{align*}<br \\\/>2\\\\pi \\\\int^1_0 (2x-x^2) dx = 2\\\\pi \\\\int^1_0 x(2-x) dx<br \\\/>\\\\end{align*}<br \\\/>\\\\)<p>\\u00a0<\\\/p><h4>First, we start with the original given integral. Then, we factor the term x out of the integrand. Since we have \\\\(2\\\\pi\\\\) at the beginning of the integral, we can safely assume that this is a shell method integral. The radius would be the factored \\\\(x\\\\), the height would be \\\\(2-x\\\\), and the limits of integration would be \\\\(0 \\\\le x \\\\le 1\\\\). The graph is as follows:<\\\/h4><p><img class=\\\"size-medium wp-image-6695 aligncenter\\\" src=\\\"https:\\\/\\\/bisonmath.com\\\/wp-content\\\/uploads\\\/2022\\\/03\\\/Screenshot-29-300x300.png\\\" alt=\\\"\\\" width=\\\"300\\\" height=\\\"300\\\" \\\/><\\\/p>\"}}]}<\/script>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-447832c0 elementor-widget elementor-widget-toggle\" data-id=\"447832c0\" data-element_type=\"widget\" data-widget_type=\"toggle.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<div class=\"elementor-toggle\">\n\t\t\t\t\t\t\t<div class=\"elementor-toggle-item\">\n\t\t\t\t\t<h2 id=\"elementor-tab-title-1141\" class=\"elementor-tab-title\" data-tab=\"1\" role=\"button\" aria-controls=\"elementor-tab-content-1141\" aria-expanded=\"false\">\n\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon elementor-toggle-icon-left\" aria-hidden=\"true\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon-closed\"><i class=\"fas fa-toggle-off\"><\/i><\/span>\n\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon-opened\"><i class=\"elementor-toggle-icon-opened fas fa-toggle-on\"><\/i><\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t\t<\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t<a class=\"elementor-toggle-title\" tabindex=\"0\">Problem 11: Solution<\/a>\n\t\t\t\t\t<\/h2>\n\n\t\t\t\t\t<div id=\"elementor-tab-content-1141\" class=\"elementor-tab-content elementor-clearfix\" data-tab=\"1\" role=\"region\" aria-labelledby=\"elementor-tab-title-1141\"><h3>Describe the solid whose volume is being represented by the given integral:<\/h3>\\(<br \/>\\begin{align*}<br \/>2\\pi \\int^1_0 (y+2)\\left(\\sqrt{1-y^2}\\right) dy<br \/>\\end{align*}<br \/>\\)<p>\u00a0<\/p><h4>Given that the integral starts with \\(2\\pi\\), we can safely assume this integral describes a solid using the shell method. The radius would be \\(y + 2\\) (meaning that the integration area is far by 2 units from the axis of rotation, let it be on the first or on the second quadrant) and the height would be described by the expression \\(\\sqrt{1 \\:- y^2}\\). Upon closer analysis, we notice that the height is the positive half of a circle of radius 1 (hence the limits of integration \\(0 \\le x \\le 1\\), yet these describe the half of a semicircle, or a quarter of a full circle). This means that this integral describes the volume of a quarter of a torus of rafius 1 with an offset of 2 units from the axis of rotation. This is a graphical representation of these explanation:<\/h4><p><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-6709 aligncenter\" src=\"https:\/\/bisonmath.com\/wp-content\/uploads\/2022\/03\/Screenshot-30-300x300.png\" alt=\"\" width=\"300\" height=\"300\" \/><\/p><\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t\t<script type=\"application\/ld+json\">{\"@context\":\"https:\\\/\\\/schema.org\",\"@type\":\"FAQPage\",\"mainEntity\":[{\"@type\":\"Question\",\"name\":\"Problem 11: Solution\",\"acceptedAnswer\":{\"@type\":\"Answer\",\"text\":\"<h3>Describe the solid whose volume is being represented by the given integral:<\\\/h3>\\\\(<br \\\/>\\\\begin{align*}<br \\\/>2\\\\pi \\\\int^1_0 (y+2)\\\\left(\\\\sqrt{1-y^2}\\\\right) dy<br \\\/>\\\\end{align*}<br \\\/>\\\\)<p>\\u00a0<\\\/p><h4>Given that the integral starts with \\\\(2\\\\pi\\\\), we can safely assume this integral describes a solid using the shell method. The radius would be \\\\(y + 2\\\\) (meaning that the integration area is far by 2 units from the axis of rotation, let it be on the first or on the second quadrant) and the height would be described by the expression \\\\(\\\\sqrt{1 \\\\:- y^2}\\\\). Upon closer analysis, we notice that the height is the positive half of a circle of radius 1 (hence the limits of integration \\\\(0 \\\\le x \\\\le 1\\\\), yet these describe the half of a semicircle, or a quarter of a full circle). This means that this integral describes the volume of a quarter of a torus of rafius 1 with an offset of 2 units from the axis of rotation. This is a graphical representation of these explanation:<\\\/h4><p><img class=\\\"size-medium wp-image-6709 aligncenter\\\" src=\\\"https:\\\/\\\/bisonmath.com\\\/wp-content\\\/uploads\\\/2022\\\/03\\\/Screenshot-30-300x300.png\\\" alt=\\\"\\\" width=\\\"300\\\" height=\\\"300\\\" \\\/><\\\/p>\"}}]}<\/script>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-fdb8651 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"fdb8651\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-7adec0e\" data-id=\"7adec0e\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-9294114 elementor-widget elementor-widget-spacer\" data-id=\"9294114\" data-element_type=\"widget\" data-widget_type=\"spacer.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<div class=\"elementor-spacer\">\n\t\t\t<div class=\"elementor-spacer-inner\"><\/div>\n\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-d68afb5 elementor-section-height-min-height elementor-section-content-middle elementor-section-stretched elementor-section-boxed elementor-section-height-default elementor-section-items-middle elementor-invisible\" data-id=\"d68afb5\" data-element_type=\"section\" 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href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/limits\/fundamentals-limits\/\" class=\"elementor-sub-item\">Fundamentals<\/a><\/li>\n\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-14279\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/limits\/limit-formulas\/\" class=\"elementor-sub-item\">Limit Formulas<\/a><\/li>\n\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-11442\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/limits\/continuity\/\" class=\"elementor-sub-item\">Continuity<\/a><\/li>\n\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-has-children menu-item-11426\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/limits\/indeterminate-limits\/\" class=\"elementor-sub-item\">Indeterminate Forms<\/a>\n\t<ul class=\"sub-menu elementor-nav-menu--dropdown\">\n\t\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-12217\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/limits\/lhopitals-rule\/\" class=\"elementor-sub-item\">L\u2019H\u00f4pital\u2019s Rule<\/a><\/li>\n\t<\/ul>\n<\/li>\n\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-12629\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/limits\/virtual-lessons-limits\/\" class=\"elementor-sub-item\">Virtual Lessons<\/a><\/li>\n<\/ul>\n<\/li>\n<li class=\"menu-item menu-item-type-custom menu-item-object-custom menu-item-has-children menu-item-14327\"><a href=\"#\" class=\"elementor-item elementor-item-anchor\">Differentiation<\/a>\n<ul class=\"sub-menu elementor-nav-menu--dropdown\">\n\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-14856\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/differentiation\/fundamentals-differentiation\/\" class=\"elementor-sub-item\">Fundamentals<\/a><\/li>\n\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-11537\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/differentiation\/differentiation-formulas-update\/\" class=\"elementor-sub-item\">Differentiation Formulas<\/a><\/li>\n\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-11409\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/differentiation\/differentiation-techniques\/\" class=\"elementor-sub-item\">Differentiation Techniques<\/a><\/li>\n\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-has-children menu-item-11411\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/differentiation\/differentiation-applications\/\" class=\"elementor-sub-item\">Differentiation Applications<\/a>\n\t<ul class=\"sub-menu elementor-nav-menu--dropdown\">\n\t\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-15049\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/limits\/lhopitals-rule\/\" class=\"elementor-sub-item\">L\u2019H\u00f4pital\u2019s Rule<\/a><\/li>\n\t\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-10229\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/differentiation\/differentiation-applications\/curve-analysis\/\" class=\"elementor-sub-item\">Curve Analysis<\/a><\/li>\n\t\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-11435\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/differentiation\/differentiation-applications\/related-rates\/\" class=\"elementor-sub-item\">Related Rates<\/a><\/li>\n\t<\/ul>\n<\/li>\n\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-20868\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/differentiation\/problem-set-differentiation\/\" class=\"elementor-sub-item\"><img decoding=\"async\" src=\"https:\/\/onlinemathtutor.co\/wp-content\/uploads\/2022\/02\/Problem-Sets.png\"  width=\"20\" style=\"margin-right: 10px;\">  Problem Set<\/a><\/li>\n<\/ul>\n<\/li>\n<li class=\"menu-item menu-item-type-custom menu-item-object-custom menu-item-has-children menu-item-14329\"><a href=\"#\" class=\"elementor-item elementor-item-anchor\">Integration<\/a>\n<ul class=\"sub-menu elementor-nav-menu--dropdown\">\n\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-has-children menu-item-14551\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/integration\/fundamentals-integration\/\" class=\"elementor-sub-item\">Fundamentals<\/a>\n\t<ul class=\"sub-menu elementor-nav-menu--dropdown\">\n\t\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-10226\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/integration\/riemann-sums\/\" class=\"elementor-sub-item\">Riemann Sums<\/a><\/li>\n\t\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-11365\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/integration\/fundamental-theorem-of-calculus\/\" class=\"elementor-sub-item\">Fundamental Theorem of Calculus<\/a><\/li>\n\t<\/ul>\n<\/li>\n\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-11530\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/integration\/integration-formulas-update\/\" class=\"elementor-sub-item\">Integration Formulas<\/a><\/li>\n\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-9815\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/integration\/integration-techinques\/\" class=\"elementor-sub-item\">Integration Techinques<\/a><\/li>\n\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-13700\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/integration\/integration-applications\/\" class=\"elementor-sub-item\">Integration Applications<\/a><\/li>\n\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-19550\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/integration\/problem-set-integration\/\" class=\"elementor-sub-item\"><img decoding=\"async\" src=\"https:\/\/onlinemathtutor.co\/wp-content\/uploads\/2022\/02\/Problem-Sets.png\"  width=\"20\" style=\"margin-right: 10px;\">  Problem Set<\/a><\/li>\n<\/ul>\n<\/li>\n<li class=\"menu-item menu-item-type-custom menu-item-object-custom menu-item-has-children menu-item-14330\"><a href=\"#\" class=\"elementor-item elementor-item-anchor\">Sequences and Series<\/a>\n<ul class=\"sub-menu elementor-nav-menu--dropdown\">\n\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-11364\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/sequences\/\" class=\"elementor-sub-item\">Fundamentals<\/a><\/li>\n\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-9804\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/sequences\/convergence-test\/\" class=\"elementor-sub-item\">Convergence Test<\/a><\/li>\n\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-10233\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/sequences\/power-series\/\" class=\"elementor-sub-item\">Power Series<\/a><\/li>\n\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-11363\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/sequences\/taylor-series\/\" class=\"elementor-sub-item\">Taylor Series<\/a><\/li>\n\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-21607\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/sequences\/problem-set-sequences-series\/\" class=\"elementor-sub-item\"><img decoding=\"async\" src=\"https:\/\/onlinemathtutor.co\/wp-content\/uploads\/2022\/02\/Problem-Sets.png\"  width=\"20\" style=\"margin-right: 10px;\">  Problem Set<\/a><\/li>\n<\/ul>\n<\/li>\n<li class=\"menu-item menu-item-type-custom menu-item-object-custom menu-item-has-children menu-item-14331\"><a href=\"#\" class=\"elementor-item elementor-item-anchor\">Parametric Equations<\/a>\n<ul class=\"sub-menu elementor-nav-menu--dropdown\">\n\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-12174\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/parametric-equations-2\/fundamentals-of-parametric-equations\/\" class=\"elementor-sub-item\">Fundamentals<\/a><\/li>\n\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-19376\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/parametric-equations-2\/polar-coordinates-calculus\/\" class=\"elementor-sub-item\">Polar Coordinates<\/a><\/li>\n\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-11366\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/precalculus-resources\/unit-circle\/\" class=\"elementor-sub-item\">Unit Circle<\/a><\/li>\n\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-12198\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/parametric-equations-2\/calculus-parametric-equation\/\" class=\"elementor-sub-item\">Calculus of Parametric Equations<\/a><\/li>\n\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-has-children menu-item-21606\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/parametric-equations-2\/problem-set-parametric-eqs-2\/\" class=\"elementor-sub-item\"><img decoding=\"async\" src=\"https:\/\/onlinemathtutor.co\/wp-content\/uploads\/2022\/02\/Problem-Sets.png\"  width=\"20\" style=\"margin-right: 10px;\">  Problem Set<\/a>\n\t<ul class=\"sub-menu elementor-nav-menu--dropdown\">\n\t\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-12615\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/parametric-equations-2\/virtual-lessons-parametric-equations\/\" class=\"elementor-sub-item\">Virtual Lessons<\/a><\/li>\n\t<\/ul>\n<\/li>\n<\/ul>\n<\/li>\n<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-10232\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/ap-calculus-frqs\/\" class=\"elementor-item\">AP Calculus Prep<\/a><\/li>\n<\/ul>\t\t\t<\/nav>\n\t\t\t\t\t<div class=\"elementor-menu-toggle\" role=\"button\" tabindex=\"0\" aria-label=\"Menu Toggle\" aria-expanded=\"false\">\n\t\t\t<i aria-hidden=\"true\" role=\"presentation\" class=\"elementor-menu-toggle__icon--open eicon-menu-bar\"><\/i><i aria-hidden=\"true\" role=\"presentation\" class=\"elementor-menu-toggle__icon--close eicon-close\"><\/i>\t\t<\/div>\n\t\t\t\t\t<nav class=\"elementor-nav-menu--dropdown elementor-nav-menu__container\" aria-hidden=\"true\">\n\t\t\t\t<ul id=\"menu-2-7f5d6b4\" class=\"elementor-nav-menu\"><li class=\"menu-item menu-item-type-custom menu-item-object-custom menu-item-14333\"><a href=\"https:\/\/onlinemathtutor.co\/index.php\/course-content\/calculus-resources\/\" class=\"elementor-item\" tabindex=\"-1\"><img decoding=\"async\" src=\"https:\/\/onlinemathtutor.co\/wp-content\/uploads\/2021\/06\/Home-Icon-Blue.png\"  width=\"20\"><\/a><\/li>\n<li class=\"menu-item menu-item-type-custom menu-item-object-custom menu-item-has-children menu-item-14328\"><a href=\"#\" class=\"elementor-item elementor-item-anchor\" tabindex=\"-1\">Limits<\/a>\n<ul class=\"sub-menu elementor-nav-menu--dropdown\">\n\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-12209\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/limits\/fundamentals-limits\/\" class=\"elementor-sub-item\" tabindex=\"-1\">Fundamentals<\/a><\/li>\n\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-14279\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/limits\/limit-formulas\/\" class=\"elementor-sub-item\" tabindex=\"-1\">Limit Formulas<\/a><\/li>\n\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-11442\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/limits\/continuity\/\" class=\"elementor-sub-item\" tabindex=\"-1\">Continuity<\/a><\/li>\n\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-has-children menu-item-11426\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/limits\/indeterminate-limits\/\" class=\"elementor-sub-item\" tabindex=\"-1\">Indeterminate Forms<\/a>\n\t<ul class=\"sub-menu elementor-nav-menu--dropdown\">\n\t\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-12217\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/limits\/lhopitals-rule\/\" class=\"elementor-sub-item\" tabindex=\"-1\">L\u2019H\u00f4pital\u2019s Rule<\/a><\/li>\n\t<\/ul>\n<\/li>\n\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-12629\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/limits\/virtual-lessons-limits\/\" class=\"elementor-sub-item\" tabindex=\"-1\">Virtual Lessons<\/a><\/li>\n<\/ul>\n<\/li>\n<li class=\"menu-item menu-item-type-custom menu-item-object-custom menu-item-has-children menu-item-14327\"><a href=\"#\" class=\"elementor-item elementor-item-anchor\" tabindex=\"-1\">Differentiation<\/a>\n<ul class=\"sub-menu elementor-nav-menu--dropdown\">\n\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-14856\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/differentiation\/fundamentals-differentiation\/\" class=\"elementor-sub-item\" tabindex=\"-1\">Fundamentals<\/a><\/li>\n\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-11537\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/differentiation\/differentiation-formulas-update\/\" class=\"elementor-sub-item\" tabindex=\"-1\">Differentiation Formulas<\/a><\/li>\n\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-11409\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/differentiation\/differentiation-techniques\/\" class=\"elementor-sub-item\" tabindex=\"-1\">Differentiation Techniques<\/a><\/li>\n\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-has-children menu-item-11411\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/differentiation\/differentiation-applications\/\" class=\"elementor-sub-item\" tabindex=\"-1\">Differentiation Applications<\/a>\n\t<ul class=\"sub-menu elementor-nav-menu--dropdown\">\n\t\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-15049\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/limits\/lhopitals-rule\/\" class=\"elementor-sub-item\" tabindex=\"-1\">L\u2019H\u00f4pital\u2019s Rule<\/a><\/li>\n\t\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-10229\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/differentiation\/differentiation-applications\/curve-analysis\/\" class=\"elementor-sub-item\" tabindex=\"-1\">Curve Analysis<\/a><\/li>\n\t\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-11435\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/differentiation\/differentiation-applications\/related-rates\/\" class=\"elementor-sub-item\" tabindex=\"-1\">Related Rates<\/a><\/li>\n\t<\/ul>\n<\/li>\n\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-20868\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/differentiation\/problem-set-differentiation\/\" class=\"elementor-sub-item\" tabindex=\"-1\"><img decoding=\"async\" src=\"https:\/\/onlinemathtutor.co\/wp-content\/uploads\/2022\/02\/Problem-Sets.png\"  width=\"20\" style=\"margin-right: 10px;\">  Problem Set<\/a><\/li>\n<\/ul>\n<\/li>\n<li class=\"menu-item menu-item-type-custom menu-item-object-custom menu-item-has-children menu-item-14329\"><a href=\"#\" class=\"elementor-item elementor-item-anchor\" tabindex=\"-1\">Integration<\/a>\n<ul class=\"sub-menu elementor-nav-menu--dropdown\">\n\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-has-children menu-item-14551\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/integration\/fundamentals-integration\/\" class=\"elementor-sub-item\" tabindex=\"-1\">Fundamentals<\/a>\n\t<ul class=\"sub-menu elementor-nav-menu--dropdown\">\n\t\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-10226\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/integration\/riemann-sums\/\" class=\"elementor-sub-item\" tabindex=\"-1\">Riemann Sums<\/a><\/li>\n\t\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-11365\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/integration\/fundamental-theorem-of-calculus\/\" class=\"elementor-sub-item\" tabindex=\"-1\">Fundamental Theorem of Calculus<\/a><\/li>\n\t<\/ul>\n<\/li>\n\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-11530\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/integration\/integration-formulas-update\/\" class=\"elementor-sub-item\" tabindex=\"-1\">Integration Formulas<\/a><\/li>\n\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-9815\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/integration\/integration-techinques\/\" class=\"elementor-sub-item\" tabindex=\"-1\">Integration Techinques<\/a><\/li>\n\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-13700\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/integration\/integration-applications\/\" class=\"elementor-sub-item\" tabindex=\"-1\">Integration Applications<\/a><\/li>\n\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-19550\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/integration\/problem-set-integration\/\" class=\"elementor-sub-item\" tabindex=\"-1\"><img decoding=\"async\" src=\"https:\/\/onlinemathtutor.co\/wp-content\/uploads\/2022\/02\/Problem-Sets.png\"  width=\"20\" style=\"margin-right: 10px;\">  Problem Set<\/a><\/li>\n<\/ul>\n<\/li>\n<li class=\"menu-item menu-item-type-custom menu-item-object-custom menu-item-has-children menu-item-14330\"><a href=\"#\" class=\"elementor-item elementor-item-anchor\" tabindex=\"-1\">Sequences and Series<\/a>\n<ul class=\"sub-menu elementor-nav-menu--dropdown\">\n\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-11364\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/sequences\/\" class=\"elementor-sub-item\" tabindex=\"-1\">Fundamentals<\/a><\/li>\n\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-9804\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/sequences\/convergence-test\/\" class=\"elementor-sub-item\" tabindex=\"-1\">Convergence Test<\/a><\/li>\n\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-10233\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/sequences\/power-series\/\" class=\"elementor-sub-item\" tabindex=\"-1\">Power Series<\/a><\/li>\n\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-11363\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/sequences\/taylor-series\/\" class=\"elementor-sub-item\" tabindex=\"-1\">Taylor Series<\/a><\/li>\n\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-21607\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/sequences\/problem-set-sequences-series\/\" class=\"elementor-sub-item\" tabindex=\"-1\"><img decoding=\"async\" src=\"https:\/\/onlinemathtutor.co\/wp-content\/uploads\/2022\/02\/Problem-Sets.png\"  width=\"20\" style=\"margin-right: 10px;\">  Problem Set<\/a><\/li>\n<\/ul>\n<\/li>\n<li class=\"menu-item menu-item-type-custom menu-item-object-custom menu-item-has-children menu-item-14331\"><a href=\"#\" class=\"elementor-item elementor-item-anchor\" tabindex=\"-1\">Parametric Equations<\/a>\n<ul class=\"sub-menu elementor-nav-menu--dropdown\">\n\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-12174\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/parametric-equations-2\/fundamentals-of-parametric-equations\/\" class=\"elementor-sub-item\" tabindex=\"-1\">Fundamentals<\/a><\/li>\n\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-19376\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/parametric-equations-2\/polar-coordinates-calculus\/\" class=\"elementor-sub-item\" tabindex=\"-1\">Polar Coordinates<\/a><\/li>\n\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-11366\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/precalculus-resources\/unit-circle\/\" class=\"elementor-sub-item\" tabindex=\"-1\">Unit Circle<\/a><\/li>\n\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-12198\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/parametric-equations-2\/calculus-parametric-equation\/\" class=\"elementor-sub-item\" tabindex=\"-1\">Calculus of Parametric Equations<\/a><\/li>\n\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-has-children menu-item-21606\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/parametric-equations-2\/problem-set-parametric-eqs-2\/\" class=\"elementor-sub-item\" tabindex=\"-1\"><img decoding=\"async\" src=\"https:\/\/onlinemathtutor.co\/wp-content\/uploads\/2022\/02\/Problem-Sets.png\"  width=\"20\" style=\"margin-right: 10px;\">  Problem Set<\/a>\n\t<ul class=\"sub-menu elementor-nav-menu--dropdown\">\n\t\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-12615\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/parametric-equations-2\/virtual-lessons-parametric-equations\/\" class=\"elementor-sub-item\" tabindex=\"-1\">Virtual Lessons<\/a><\/li>\n\t<\/ul>\n<\/li>\n<\/ul>\n<\/li>\n<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-10232\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/ap-calculus-frqs\/\" class=\"elementor-item\" tabindex=\"-1\">AP Calculus Prep<\/a><\/li>\n<\/ul>\t\t\t<\/nav>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t<div class=\"elementor-column elementor-col-50 elementor-top-column elementor-element elementor-element-1c1e1cd\" data-id=\"1c1e1cd\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-788e7a3 elementor-widget elementor-widget-spacer\" data-id=\"788e7a3\" data-element_type=\"widget\" data-widget_type=\"spacer.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<div class=\"elementor-spacer\">\n\t\t\t<div class=\"elementor-spacer-inner\"><\/div>\n\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<\/div>\n\t\t","protected":false},"excerpt":{"rendered":"<p>Practice Problems: Rotational Volumes To determine the volume of rotational symmetries there are two main protocols to follow: the Washer\/Disk Method or the Shell Method. The washer method is a generalization of the disk method. Begin by sketching a 2D <a href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/integration\/problem-set-integration\/integration-practice-problems-application-rotational-volumes\/\" class=\"read-more\">Read More &#8230;<\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"parent":19541,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"elementor_header_footer","meta":{"footnotes":""},"class_list":["post-19867","page","type-page","status-publish","hentry"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v24.6 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>Integration Practice Problems - [Application: Rotational Volumes] | Online Math Tutoring LLC | Expert Math Tutor Near Me<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/onlinemathtutor.co\/course-content\/calculus-resources\/integration\/problem-set-integration\/integration-practice-problems-application-rotational-volumes\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Integration Practice Problems - [Application: Rotational Volumes] | Online Math Tutoring LLC | Expert Math Tutor Near Me\" \/>\n<meta property=\"og:description\" content=\"Practice Problems: Rotational Volumes To determine the volume of rotational symmetries there are two main protocols to follow: the Washer\/Disk Method or the Shell Method. 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