Exercises

1. Approximate the integral ${\displaystyle \underset{0}{\overset{\pi }{\int }}x\sin \left(x\right)dx}$ using Simpson’s Method with N=8. Judging from the result that you got, what do you think the value of this integral probably is? You can check this using Maple to get the analytical result.

   f(0)	0
   f( $\pi$ /8)	0.1502794
   f( $\pi$ /4)	0.5553608
   f(3 $\pi$ /8)	1.088419
   f( $\pi$ /2)	1.570796
   f(5 $\pi$ /8)	1.814033
   f(3 $\pi$ /4)	1.666081
   f(7 $\pi$ /8)	1.051956
   f( $\pi$ )	0

   $\begin{array}{l}{\displaystyle \underset{0}{\overset{\pi }{\int }}x\sin \left(x\right)dx}\approx \frac{\pi /8}{3}\left[\begin{array}{l}0+4\left(\text{0}\text{.1502794}\right)+2\left(\text{0}\text{.5553608}\right)+4\left(\text{1}\text{.088419}\right)+2\left(\text{1}\text{.570796}\right)\\ +4\left(\text{1}\text{.814033}\right)+2\left(\text{1}\text{.666081}\right)+4\left(\text{1}\text{.051956}\right)+0\end{array}\right]\\ {\displaystyle \underset{0}{\overset{\pi }{\int }}x\sin \left(x\right)dx}\approx 3.142014\end{array}$