Chapter 5.9, Problem 6E

a.

To determine

To calculate: The given integral with the specified value of n.

Answer to Problem 6E

approximate the given integral ${\displaystyle \underset{0}{\overset{\pi }{\int }}x\cos xdx}\approx -1.945744$ .

Explanation of Solution

Given information: Use a Midpoint Rule to approximate the given integral with the specified value of n.

  ${\displaystyle \underset{0}{\overset{\pi }{\int }}x\cos xdx,n=4}$

Formula used: Midpoint Rule

  $\begin{array}{l}{\displaystyle \underset{a}{\overset{b}{\int }}f(x)dx\approx \Delta x}\left[f\left(\overline{x_1}\right)+f\left(\overline{x_2}\right)+\cdot \cdot \cdot \cdot \cdot +f\left(\overline{x_{n-1}}\right)+f\left(\overline{x_n}\right)\right]\\ \text{Where }\Delta x=\frac{b-a}{n}\text{ and }\overline{x_i}=\frac{\left(x_{i-1}+x_i\right)}{2}=\text{ midpoint of }\left[x_{i-1},x_i\right].\end{array}$

Calculation: Use a Midpoint Rule to approximate the given integral with the specified value of n.

  $\begin{array}{l}{\displaystyle \underset{0}{\overset{\pi }{\int }}x\cos xdx,n=4}\\ {\displaystyle \underset{0}{\overset{\pi }{\int }}x\cos xdx}\approx \Delta x\left[f\left(\overline{x_1}\right)+f\left(\overline{x_2}\right)+\cdot \cdot \cdot \cdot \cdot +f\left(\overline{x_{n-1}}\right)+f\left(\overline{x_n}\right)\right]\\ \Delta x=\frac{\pi -0}{4}\\ \Delta x=\frac{\pi }{4}\\ {\displaystyle \underset{0}{\overset{\pi }{\int }}x\cos xdx}\approx \frac{\pi }{4}\left[0.362806+0.450838-0.751397-2.539646\right]\\ {\displaystyle \underset{0}{\overset{\pi }{\int }}x\cos xdx}\approx -1.945744\\ \text{And}\\ {\displaystyle \underset{0}{\overset{\pi }{\int }}x\cos xdx}={\left[x\sin x\right]}_0^{\pi }-{\displaystyle \underset{0}{\overset{\pi }{\int }}\sin xdx}\\ {\displaystyle \underset{0}{\overset{\pi }{\int }}x\cos xdx}={\left[x\sin x\right]}_0^{\pi }-{\left[-\cos x\right]}_0^{\pi }\\ {\displaystyle \underset{0}{\overset{\pi }{\int }}x\cos xdx}=-2\end{array}$

The error of the approximation is about

  $-2+1.945744\approx -0.054256$

b.

To determine

To calculate: The given integral with the specified value of n.

Answer to Problem 6E

approximate the given integral ${\displaystyle \underset{0}{\overset{\pi }{\int }}x\cos xdx}\approx -1.985611$ .

Explanation of Solution

Given information: Use a Simpson’s Rule to approximate the given integral with the specified value of n.

  ${\displaystyle \underset{0}{\overset{\pi }{\int }}x\cos xdx,n=4}$

Formula used: Simpson’s Rule

  $\begin{array}{l}{\displaystyle \underset{a}{\overset{b}{\int }}f(x)dx\approx \frac{\Delta x}{3}}\left[f\left(x_0\right)+4f\left(x_1\right)+2f\left(x_2\right)+\cdot \cdot \cdot \cdot \cdot +4f\left(x_{n-1}\right)+f\left(x_n\right)\right]\\ \text{Where }n\text{ is even and }\Delta x=\frac{b-a}{n}.\end{array}$

Calculation: Use a Simpson’s Rule to approximate the given integral with the specified value of n.

  $\begin{array}{l}{\displaystyle \underset{0}{\overset{\pi }{\int }}x\cos xdx,n=4}\\ \Delta x=\frac{\pi -0}{4}\\ \Delta x=\frac{\pi }{4}\\ x_0=0.000000\\ x_1=0.785398\\ x_2=1.570796\\ x_3=2.356194\\ x_4=3.141592\\ f\left(x_0\right)=0.000000\\ f\left(x_1\right)=0.555360\\ f\left(x_2\right)=0.000000\\ f\left(x_3\right)=-1.666081\\ f\left(x_4\right)=-3.141592\\ {\displaystyle \underset{0}{\overset{\pi }{\int }}x\cos xdx}\approx \frac{\Delta x}{3}\left[f\left(x_0\right)+4f\left(x_1\right)+2f\left(x_2\right)+4f\left(x_3\right)+f\left(x_4\right)\right]\\ {\displaystyle \underset{0}{\overset{\pi }{\int }}x\cos xdx}\approx -1.985611\\ \text{And}\\ {\displaystyle \underset{0}{\overset{\pi }{\int }}x\cos xdx}=-2\end{array}$

The error of the approximation is about

  $-2+1.985611\approx -0.014389$