The Left-Hand Rule (LHR) is a rudimentary numerical integration technique for approximating the area under the curve. This technique combines (integrate) the areas of rectangular regions. It is common, but not necessary, to first subdivide a given interval \([a,b]\) into \(n\) equally spaced subintervals whose length is typically denoted as \(\textstyle\Delta x=\frac{b-a}{n}\).

The height for each rectangular region is \(f(x_k)\) where \(x_k\) is the left-hand endpoint on the interval, thus, the reason behind the mystery why this rule is known as the Left-Hand Rule.

\(\displaystyle\texttt{Area}\approx \sum_{k=0}^{n-1}f(x_k)\Delta x\)

where \(x_k=a+k\Delta x\) for \(k=0,1,2,\dots,n\).

Notice that \(a=x_0\) and \(b=x_n\)

The Right-Hand Rule (RHR) is a rudimentary numerical integration technique for approximating the area under the curve. This technique combines (integrate) the areas of rectangular regions. It is common, but not necessary, to first subdivide a given interval \([a,b]\) into \(n\) equally spaced subintervals whose length is typically denoted as \(\textstyle\Delta x=\frac{b-a}{n}\).

The height for each rectangular region is \(f(x_{k+1})\) where \(x_{k+1}\) is the right-hand endpoint on the interval, thereby, once again demystifying the mystery why this rule is known as the Right-Hand Rule.

\(\displaystyle\texttt{Area}\approx\sum_{k=1}^{n}f(x_k)\Delta x\)

where \(x_k=a+k\Delta x\) for \(k=0,1,2,\dots,n\).

Notice that \(a=x_0\) and \(b=x_n\)

The Midpoint Rule is a numerical integration technique that seeks to balance the error of over-approximating and under-approximating found within the Left-Hand and the Right-Hand Rule.

\(\displaystyle\sum_{k=1}^{n}f(\overline{x}_k)\Delta x\)  where  \(\displaystyle\overline{x}_k=\frac{x_k+x_{k-1}}{2}\)  is the midpoint of the interval \([x_{k+1},x_k]\)

NOTE: The relationship between the Midpoint Rule, Trapezoidal Rule, and Simpson’s Rule is given by the following identity. \(\displaystyle S_{2n}=\frac{T_n+2M_n}{3}\)

Error Term

\(\displaystyle|E_T|\leq\frac{M_2(b-a)^3}{24n^4}\) where \(M_2\geq|f^{(2)}(x)\) for \(a\leq x\leq b\)

The Trapezoidal Rule is a numerical integration technique that approximates the area under the curve by integrating the areas of various trapezoids. The number of partitions \(n\) may either be even or odd.

\(\displaystyle T_n=\frac{b-a}{2n}\big(f(x_0)+2f(x_1)+\cdots+2f(x_{n+-1})+f(x_n) \big)\)

\(\displaystyle \Delta x=\frac{b-a}{n}\quad
\displaystyle x_k=a+k\Delta x\)

Error Term

\(\displaystyle|E_T|\leq\frac{M_2(b-a)^3}{12n^4}\) where \(M_2\geq|f^{(2)}(x)\) for \(a\leq x\leq b\)

Also known as Simpson’s \(\textstyle\frac{1}{3}\) Rule is a numerical integration technique that improves upon the Trapezoidal Rule by utilizing the geometry of parabolic arcs. The number of partitions \(n\) must be even.

\(\displaystyle S_n=\frac{b-a}{3n}\big( f(x_0)+4f(x_1)+2f(x_2)+\cdots+4f(x_{n-1})+2f(x_n) \big)\)

\(\displaystyle \Delta x=\frac{b-a}{n}\quad
\displaystyle x_k=a+k\Delta x\)

Error Term

\(\displaystyle|E_S|\leq\frac{M_4(b-a)^5}{180n^4}\) where \(M_4\geq|f^{(4)}(x)\) for \(a\leq x\leq b\)