Example 11. Find the volume of the body bounded by two cylinders $x^{2}+y^{2}=R^{2}$ and $x^{2}+z^{2}=R^{2}$.
Solution. In Fig. 2.11, one-eighth of the body located in the first octant $(x \geqslant 0, y \geqslant 0, z \geqslant 0)$ is shown. The cross-section by a plane perpendicular to the $O x$ axis is a square. If the section is made through the point with abscissa $(x, 0,0)$, then the side of the square is $a=y=z=\sqrt{R^{2}-x^{2}}$, and its area is $S(x)=a^{2}=R^{2}-x^{2}, 0 \leqslant x \leqslant R$.

![](https://cdn.mathpix.com/cropped/2024_05_22_446938f1e4808e7d6e07g-040.jpg?height=578&width=712&top_left_y=789&top_left_x=284)

Fig. 2.11

Using the appropriate formula (6), we get:

$$
\frac{1}{8} V=\int_{0}^{R}\left(R^{2}-x^{2}\right) d x=\left.\left(R^{2} x-\frac{x^{3}}{3}\right)\right|_{0} ^{R}=\frac{2}{3} R^{3}
$$

Answer. $V=\frac{16}{3} R^{3}$.