Simpson's Rule

The Simpson's rule approximation Sn (n an even positive integer) is given for an arbitrary function f by

$\displaystyle \int_{{a}}^{{b}}$f (x) dx $\displaystyle \approx$ Sn  
  = $\displaystyle {\frac{{b-a}}{{3n}}}$$\displaystyle \left(\vphantom{ f\left( a\right) +f\left( b\right)
+4\sum_{i=1}^{n/2}f\left( a+\left( 2i-1\right) \frac{b-a}{n}\right) }\right.$f$\displaystyle \left(\vphantom{ a}\right.$a$\displaystyle \left.\vphantom{ a}\right)$ + f$\displaystyle \left(\vphantom{ b}\right.$b$\displaystyle \left.\vphantom{ b}\right)$ +4$\displaystyle \sum_{{i=1}}^{{n/2}}$f$\displaystyle \left(\vphantom{ a+\left( 2i-1\right) \frac{b-a}{n}}\right.$a + $\displaystyle \left(\vphantom{ 2i-1}\right.$2i - 1$\displaystyle \left.\vphantom{ 2i-1}\right)$$\displaystyle {\frac{{b-a}}{{n}}}$$\displaystyle \left.\vphantom{ a+\left( 2i-1\right) \frac{b-a}{n}}\right)$  
    $\displaystyle \left.\vphantom{ +2\sum_{i=1}^{-1+n/2}f\left( a+2i\frac{b-a}{n}\right) }\right.$ +2$\displaystyle \sum_{{i=1}}^{{-1+n/2}}$f$\displaystyle \left(\vphantom{ a+2i\frac{b-a}{n}}\right.$a + 2i$\displaystyle {\frac{{b-a}}{{n}}}$$\displaystyle \left.\vphantom{ a+2i\frac{b-a}{n}}\right)$$\displaystyle \left.\vphantom{ +2\sum_{i=1}^{-1+n/2}f\left( a+2i\frac{b-a}{n}\right) }\right)$  

The error bound for Simpson's rule is given by

$\displaystyle \left\vert\vphantom{ S_{n}-\int_{a}^{b}f(x)\,dx}\right.$Sn - $\displaystyle \int_{{a}}^{{b}}$f (x) dx$\displaystyle \left.\vphantom{ S_{n}-\int_{a}^{b}f(x)\,dx}\right\vert$K$\displaystyle {\frac{{(b-a)^{5}}}{{180n^{4}}%
}}$


where K is any number such that $\left\vert\vphantom{ f^{(4)}(x)}\right.$f(4)(x)$\left.\vphantom{ f^{(4)}(x)}\right\vert$K for all x$\left[\vphantom{ a,b}\right.$a, b$\left.\vphantom{ a,b}\right]$. In particular, Simpson's rule is exact for integrals of polynomials of degree at most 3 (because the fourth derivative of such a polynomial is identically zero).

$\blacktriangleright$ To approximate $\int_{{0}}^{{\pi }}$x sin x dx using Simpson's rule

1.
Leave the insertion point in the expression $\int_{{0}}^{{\pi }}$x sin x dx.

2.
From the Calculus submenu, choose Approximate Integral.

3.
Choose Simpson in the dialog box with 10 Subintervals.

$\blacktriangleright$ Calculus + Approximate Integral + Simpson

$\dint_{{0}}^{{\pi }}$x sin x dx Approximate integral (Simpson's rule) is

${\frac{{1}}{{30}}}$π$\left(\vphantom{ 4\dsum\limits_{i=1}^{5}\frac{1}{10}\left(
2i-1\right) \pi \si...
...ght) \pi
+2\dsum\limits_{i=1}^{4}\frac{1}{5}i\pi \sin \frac{1}{5}i\pi }\right.$4$\dsum\limits_{{i=1}}^{{5}}$${\frac{{1}}{{10}}}$$\left(\vphantom{
2i-1}\right.$2i - 1$\left.\vphantom{
2i-1}\right)$πsin${\frac{{1}}{{10}}}$$\left(\vphantom{
2i-1}\right.$2i - 1$\left.\vphantom{
2i-1}\right)$π +2$\dsum\limits_{{i=1}}^{{4}}$${\frac{{1}}{{5}}}$sin${\frac{{1}}{{5}}}$$\left.\vphantom{ 4\dsum\limits_{i=1}^{5}\frac{1}{10}\left(
2i-1\right) \pi \si...
...ght) \pi
+2\dsum\limits_{i=1}^{4}\frac{1}{5}i\pi \sin \frac{1}{5}i\pi }\right)$

$\blacktriangleright$ Evaluate Numerically

${\frac{{1}}{{30}}}$π$\left(\vphantom{ 4\dsum\limits_{i=1}^{5}\frac{1}{10}\left(
2i-1\right) \pi \si...
...ght) \pi
+2\dsum\limits_{i=1}^{4}\frac{1}{5}i\pi \sin \frac{1}{5}i\pi }\right.$4$\dsum\limits_{{i=1}}^{{5}}$${\frac{{1}}{{10}}}$$\left(\vphantom{
2i-1}\right.$2i - 1$\left.\vphantom{
2i-1}\right)$πsin${\frac{{1}}{{10}}}$$\left(\vphantom{
2i-1}\right.$2i - 1$\left.\vphantom{
2i-1}\right)$π +2$\dsum\limits_{{i=1}}^{{4}}$${\frac{{1}}{{5}}}$sin${\frac{{1}}{{5}}}$$\left.\vphantom{ 4\dsum\limits_{i=1}^{5}\frac{1}{10}\left(
2i-1\right) \pi \si...
...ght) \pi
+2\dsum\limits_{i=1}^{4}\frac{1}{5}i\pi \sin \frac{1}{5}i\pi }\right)$ = 3.1418



\begin{example}
To find the number of subdivisions required to approximate $%
...
...^{-x^{2}}-48x^{2}e^{-x^{2}}+16x^{4}e^{-x^{2}}
\end{displaymath}
\end{example}

$\blacktriangleright$ Plot 2D + Rectangular

12e-x2 -48x2 +16x4e-x2

dtbpF3in2.0003in0pt


\begin{example}
From the graph, you can see that $f^{(4)}(x)$\ has a maximum va...
...vert =8.155\times
10^{-7}<10^{-5}
\end{displaymath}
\medskip
\end{example}