1308. Calculate the volume of the body obtained by rotating around the $O x$ axis the figure bounded by the parabolas $y=x^{2}$ and $x=y^{2}$.
Solution. Solving the system of equations

$$
\left\{\begin{aligned}
y & =x^{2} \\
y^{2} & =x
\end{aligned}\right.
$$

we get \(x_{1}=0, x_{2}=1, y_{1}=0, y_{2}=1\), from which the points of intersection of the curves are \(O(0 ; 0)\) and \(B(1 ; 1)\). As can be seen (Fig. 88), the volume of the solid of revolution is the difference between two volumes formed by rotating around the \(O x\) axis the curvilinear trapezoids \(O C B A\) and \(O D B A\):

$$
\begin{aligned}
V=V_{1}-V_{2}= & \pi \int_{0}^{1} x d x-\pi \int_{0}^{1} x^{4} d x=\left.\pi\left(\frac{x^{2}}{2}-\frac{x^{5}}{5}\right)\right|_{0} ^{1}= \\
& =\pi\left(\frac{1}{2}-\frac{1}{5}\right)=\frac{3}{10} \pi
\end{aligned}
$$