Answer
$\displaystyle{V=\frac{\pi^2}{4}}$
Work Step by Step
$\displaystyle{A(x)=\pi\left(\sin x\right)^2}\\
\displaystyle{A(x)=\pi\left(\sin^2 x\right)}\\$
$\displaystyle{V=\int_{\frac{\pi}{2}}^{\pi}A(x)\ dx}\\
\displaystyle{V=\int_{\frac{\pi}{2}}^{\pi}\pi\left(\sin^2 x\right)\ dx}\\
\displaystyle{V=\pi\int_{\frac{\pi}{2}}^{\pi}\frac{1-\cos2x}{2}\ dx}\\
\displaystyle{V=\frac{\pi}{2}\int_{\frac{\pi}{2}}^{\pi}1-\cos2x\ dx}\\
\displaystyle{V=\frac{\pi}{2}\left[x-\frac{1}{2}\sin2x\right]_{\frac{\pi}{2}}^{\pi}}\\
\displaystyle{V=\frac{\pi^2}{4}}$
