Example 5. Calculate the moment of inertia of a right circular cylinder of radius $R$ and height $H$ about a diameter of its middle section, if the density is constant and equal to $k$.
Solution. Let's choose a coordinate system such that the $O z$ axis is directed along the axis of the cylinder, and the plane $O x y$ passes through its center. We need to compute $I_{x}=\iint_{W} \int\left(y^{2}+z\right) \cdot k d x d y d z$.

We switch to cylindrical coordinates (omitting some details):

$$
\begin{aligned}
& \begin{aligned}
& I_{x}=k \int_{0}^{2 \pi} d \varphi \int_{0}^{R} \int_{-H}^{H}\left(z^{2}+r^{2} \sin ^{2} \varphi\right) r d r d z= \\
&=k \int_{0}^{2 \pi} d \varphi \int_{0}^{R}\left(\frac{2 H^{3}}{3}+2 H r^{2} \sin ^{2} \varphi\right) r d r= \\
&=k \int_{0}^{2 \pi}\left(\frac{2 H^{3}}{3} \cdot \frac{R^{2}}{2}+\frac{2 H R^{4}}{4} \sin ^{2} \varphi\right) d \varphi=k \pi H R^{2}\left(\frac{2}{3} H^{2}+\frac{R^{2}}{2}\right)
\end{aligned} \\
& \text { Answer. } I_{x}=k \pi H R^{2}\left(\frac{2}{3} H^{2}+\frac{R^{2}}{2}\right)
\end{aligned}
$$