Volume of the solid of revolution 

Using the disk method, the volume is given by
\[\int_{0}^{\pi}\pi \sin^2x dx=\int_0^{\pi}\pi (\frac{1-\cos 2x}{2})dx\]
\[=\frac{\pi}{2} (x-\frac{\sin 2x}{2})\big|_0^{\pi}\]
\[=\frac{\pi}{2} (\pi-\frac{\sin 2\pi}{2})-\frac{\pi}{2} (0-\frac{\sin 0}{2})\]
\[=\frac{\pi}{2} (\pi-0)-\frac{\pi}{2} (0-0)=\frac{\pi^2}{2}.\]