Chapter 8.9, Problem 17E

To determine

Calculate $S_N$ given by Simpson’s rule for the value of $N$ indicated.

$S_4=0.7469$

Given information:

${\displaystyle {\int }_0^1{e}^{-x^2}dx,N=4}$

Formula used:

For $N$ even the $N^{th}$ approximation ${\displaystyle {\int }_a^{b}f\left(x\right)dx}$ by Simpson’s rule is

$S_N=\frac{1}{3}\Delta x\left[y_0+2\left(y_2+y_4+\mathrm{....}+y_{N-2}\right)+4\left(y_1+y_3+\mathrm{....}+y_{N-1}\right)+y_N\right]$

Where

$\Delta x=\frac{b-a}{N}$

And

$y_j=f\left(a+j\Delta x\right)$

Calculation:

First to find $\Delta x$:

Recollect $\Delta x=\frac{b-a}{N}$

Substitute $a=0,b=1\text{ and }N=4$

$\begin{array}{c}\Delta x=\frac{1-0}{4}\\ =\frac{1}{4}\end{array}$

And to find $y_j=f\left(a+j\Delta x\right)\text{ for }j=0,1,2,\mathrm{........}$

$\begin{array}{c}{y}_0=f\left(0+0\times \frac{1}{4}\right)\\ =f\left(0\right)\\ =1\\ y_1=f\left(0+1\times \frac{1}{4}\right)\\ =f\left(\frac{1}{4}\right)\\ =0.99961\end{array}$

Find the reaming values as follows:

$\begin{array}{c}{y}_2=f\left(\frac{2}{4}\right)\\ =09845\\ y_3=f\left(\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$4$}\right.\right)\\ =0.9655\\ y_4=f\left(\raisebox{1ex}{$4$}\!\left/ \!\raisebox{-1ex}{$4$}\right.\right)\\ =0.9394\end{array}$

Substitute all values in the formula of $S_N$.

$\begin{array}{c}{S}_N=\frac{1}{3}\Delta x\left[y_0+2\left(y_2+y_4+\mathrm{.......}+y_{N-2}\right)+4\left(y_1+y_3+\mathrm{......}+y_{N-1}\right)+y_N\right]\\ S_4=\frac{1}{3}\left(\frac{1}{4}\right)\left[1+4\times 0.9961+2\times 0.9845+4\times 0.9655+0.9394\right]\\ =\frac{1}{12}\left[1+3.9844+1.969+3.862+0.9394\right]\\ =\frac{11.7548}{12}\\ =0.7469\end{array}$

Therefore,

$S_4=0.7469$