Chapter 8, Problem 19RE

To determine

To find : the area of the region enclosed by the lines and curves.

Answer to Problem 19RE

The area of the region is $4\pi$ .

Explanation of Solution

Given information : the region enclosed between the graphs of $y=x\sin x,y=-x\sin x$ over the interval $\left[-\pi ,\pi \right]$ .

Calculation :

The function $y=x\sin x,y=-x\sin x$ .

The integral is intersects at $x=\pm \pi$ .

The integral to find the area bounded by these curve:

  $\begin{array}{l}{\displaystyle \underset{x}{\overset{-x}{\int }}\left(xisnx-\left(-x\sin x\right)\right)dx}\\ {\displaystyle \underset{-\pi }{\overset{\pi }{\int }}2x\sin xdx}\\ \Rightarrow 2\sin \left(x\right)-2x\cos \left(x\right)+C\\ \Rightarrow 2\left(\sin \left(x\right)\right)-x\cos \left(x\right)+C\\ \Rightarrow 4\pi \end{array}$