Chapter 8, Problem 24RE

To determine

To find : the volume of the solid.

Answer to Problem 24RE

The volume of the solid is $\frac{{\pi }^2}{4}$ .

Explanation of Solution

Given information : the base of a solid is the region enclosed between the graphs of $y=\sin x$ and $y=-\sin x$ from $x=0$ to $x=\pi$ . Each cross section perpendicular to the x -axis is a semicircle with diameter connecting the two graphs.

Calculation :

The volume generated solid is:

  $\begin{array}{l}{\displaystyle \underset{0}{\overset{\pi }{\int }}\pi \frac{{\sin }^2\left(x\right)}{2}}dx\\ \Rightarrow \frac{\pi \left(\sin 2x\right)-2x}{8}\\ \Rightarrow \frac{\pi \left(x-\frac{\sin \left(2x\right)}{2}\right)}{4}\\ \Rightarrow \frac{{\pi }^2}{4}\end{array}$