Example 3.19. Calculate the volume of the body obtained by rotating around the Ox axis the figure bounded by the parabolas $y=x^{2}$ and $x=y^{2}$ (Fig. 43).

![](https://cdn.mathpix.com/cropped/2024_05_22_e915d089cc29fd681be5g-168.jpg?height=632&width=857&top_left_y=170&top_left_x=298)

Fig. 43
Solution. Solving the system of equations

$$
\begin{aligned}
& y=x^{2} \\
& y^{2}=x, x^{4}=x, x^{4}-x=0, x(x-1)\left(x^{2}+x+1\right)=0
\end{aligned}
$$

we get $x_{1}=0, x_{2}=1, y_{1}=0, y_{2}=1$, from which the points of intersection of the curves are $O(0 ; 0), B(1 ; 1)$. As can be seen (Fig. 43), 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 $O D B A$:

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

## 3.3. Functions of Several Variables