Integration Applications

Average Value
of a Function
Integration

The average value of the function \(f(x)\) over the interval \((a,b)\), can be computed by evaluating the given integral.

\(\displaystyle\frac{1}{b-a}\int_a^bf(x)dx\)

Area Between Curves
Integration

If \(g(x)\leq f(x)\) for over the entire interval \((a,b)\), then the area between these two curves can be computed by evaluating the following integral

\(\displaystyle\int_a^b\big(f(x)-g(x)\big)dx\)

Computing Areas

By extending the ideas that lead to the idea of Riemann Sums allows us to interpret a definite integral as the “Area Under the Curve.” If the integrated \(f(x)\) is above the x-axis then the area is positive, whereas, if \(f(x)\) is below the x-axis then the area is negative.

\(\texttt{Area Under The Curve}=\displaystyle\int_a^b f(x) dx \)

Computing Volumes

The volume for an object can be computed by taking slices, known as cross-sections, whose volume is infinitely small. We then include/combine/combine/integrate these areas together to form the object. Therefore,

\(\texttt{Volume}=\displaystyle\int\texttt{Area}\)

Special Case: Rotational Symmetries

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 DO NOT 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.

To compute the radius, always measure FROM (not TO but FROM) the axis of the rotation.

Washer/Disk Method
- Cross-Sections are taken perpendicular to the Axis of Rotation
(Cylindrical) Shell Method
- Cross-Sections are taken parallel to the Axis of Rotation

Volume By Slices
Integration

To determine the volume of non-rotational objects, consider taking adavanteous cross-section so that the slices are congruent shapes. To solve these problems, begin by sketching out both a 2D and 3D model so that you can better understand the probelm.

\(\texttt{Volume}=\displaystyle\int\texttt{Area}\)

Washer Method
Integration

Cross-sections are takan perpendicular to the axis of rotation. \(R(x)\) is the outside radius, whereas, \(r(x)\) is the inside radius.

\(\displaystyle \pi\int_{a}^{b}\big( [R(x)]^2-[r(x)]^2\big)dx \)

Disk Method
Integration

Cross-sections are taken perpendicular to the axis of rotation. To compute the radius, always measure FROM (not TO but FROM) the axis of the rotation.

\(\displaystyle \pi\int_{a}^{b}\big([R(x)]^2\big)dx \)

Shell Method
Integration

Cross-sections are taken parallel to the axis of rotation. To compute the radius, always measure FROM (not TO but FROM) the axis of the rotation.

\(\displaystyle 2\pi\int_{a}^{b}\big(\texttt{height}\big)\big(\texttt{radius}\big)dx \)

Arc Length
Integration

The integral is based upon approximating the length of line segments by using the Pythagorean Theorem.

\(\displaystyle \int_{a}^{b}\sqrt{1+(f'(x))^2}dx \)

Surface Area
Integration

\(2\pi \displaystyle \int_{a}^{b}f(x)\sqrt{1+(f'(x))^2}dx \)

\(2\pi \displaystyle \int_{a}^{b}f(y)\sqrt{1+(g'(y))^2}dy \)