Example 4. Calculate the moments of inertia of the circle $x^{2}+y^{2} \leqslant-x$ with respect to the coordinate axes and the origin. Assume the density of the circle is equal to one.
Solution. We will transition to polar coordinates. The circle $x^{2}+y^{2} \leqslant -x$, i.e., $\left(x+\frac{1}{2}\right)^{2}+y^{2} \leqslant \frac{1}{4}$, is located in the second and third quadrants, so $\frac{\pi}{2} \leqslant \varphi \leqslant \frac{3 \pi}{2}$. In this case, $0 \leqslant r \leqslant -\cos \varphi$ (where $r = -r \cos \varphi$ is the polar equation of the circle). We will compute $I_{O x}$ and $I_{O y}$ separately.

\[
\begin{aligned}
I_{O x}=\iint_{D} y^{2} d x d y= & \int_{\pi / 2}^{3 \pi / 2} \sin ^{2} \varphi \int_{0}^{-\cos \varphi} r^{3} d r=\frac{1}{4} \int_{\pi / 2}^{3 \pi / 2} \sin ^{2} \varphi \cos ^{4} \varphi d \varphi= \\
= & 2 \cdot \frac{1}{4} \int_{\pi / 2}^{\pi} \frac{1}{4} \sin ^{2} 2 \varphi \cdot \frac{1+\cos 2 \varphi}{2} d \varphi= \\
& =\frac{1}{16}\left(\int_{\pi / 2}^{\pi} \sin ^{2} 2 \varphi d \varphi+\int_{\pi / 2}^{\pi} \sin ^{2} 2 \varphi \cos 2 \varphi d \varphi\right)=\frac{\pi}{64}
\end{aligned}
\]

\[
\begin{aligned}
& I_{O y}=\int_{D} x^{2} d x d y=\int_{\pi / 2}^{3 \pi / 2} \cos ^{2} \varphi \int_{0}^{-\cos \varphi} r^{3} d r=\frac{1}{4} \int_{\pi / 2}^{3 \pi / 2} \cos ^{6} \varphi d \varphi= \\
& =\frac{1}{2} \int_{\pi / 2}^{\pi}\left(\frac{1+\cos 2 \varphi}{2}\right)^{3} d \varphi= \\
& =\frac{1}{16} \int_{\pi / 2}^{\pi}\left[1+3 \cos 2 \varphi+3 \cdot \frac{1+\cos 4 \varphi}{2}+\left(1-\sin ^{2} 2 \varphi\right) \cos 2 \varphi\right] d \varphi=\frac{5}{64} \pi \\
& I_{O}=\int_{D} \int_{D}\left(x^{2}+y^{2}\right) d x d y=\int_{\pi / 2}^{\pi} d \varphi \int_{0}^{\cos \varphi} r^{3} d r=2 \int_{\pi / 2}^{\pi} \frac{\cos ^{4} \varphi}{4} d \varphi= \\
& =\frac{1}{2} \int_{\pi / 2}^{\pi} \frac{(1+\cos 2 \varphi)^{2}}{4} d \varphi=\frac{1}{8} \int_{\pi / 2}^{\pi}\left[1+2 \cos 2 \varphi+\frac{1}{2}(1+2 \cos 4 \varphi) d \varphi\right]=\frac{3 \pi}{32} \\
& \text { Answer. } I_{O x}=\frac{\pi}{64}, I_{O y}=\frac{5 \pi}{64}, I_{O}=\frac{3 \pi}{32}
\end{aligned}
\]