Example 2.71. Find the moment of inertia of the arc of the parabola $y=x^{2}(0 \leqslant x \leqslant 1)$ with respect to the origin, if the linear density $\rho=\rho_{0} \sqrt{1+4 x}$.
Solution. We will use formula (2.48):

$$
\begin{aligned}
I_{0}= & \int_{0}^{1} \rho_{0} \sqrt{1+4 x}\left(x^{2}-\left(x^{2}\right)^{2}\right) \sqrt{1+(2 x)^{2}} d x= \\
& =\rho_{0} \int_{0}^{1}\left(x^{2}+4 x^{3}+x^{4}+4 x^{5}\right) d x= \\
= & \left.\rho_{0}\left(\frac{x^{3}}{3}+4 x^{3}+\frac{x^{5}}{5}+\frac{2 x^{6}}{3}\right)\right|_{0} ^{1}=2.2 \rho_{0}
\end{aligned}
$$