Example 3.18. Calculate the volume of the body formed by rotating the figure bounded by one half-wave of the sine curve $y=\sin x$ around the $O x$ axis.
## Solution.

$$
\begin{gathered}
V=\pi \int_{0}^{\pi} y^{2} d x=\pi \int_{0}^{\pi} \sin ^{2} x d x=\pi \int_{0}^{\pi} \frac{1-\cos 2 x}{2} d x=\frac{\pi}{2} \int_{0}^{\pi}(1-\cos 2 x) d x= \\
=\frac{\pi}{2} x-\left.\frac{1}{2} \sin 2 x\right|_{0} ^{\pi}=\frac{\pi^{2}}{2} .
\end{gathered}
$$