621. Find the volume common to two cylinders: $x^{2}+y^{2}=a^{2}$ and $y^{2}+z^{2}=a^{2}$ (bounded by these cylindrical surfaces).
Solution. We will construct one-eighth of the body located in the first octant, see Fig. 99.

Any section of the body by a plane parallel to the $x O z$ plane is a square. The area of the section $P Q N M$, at a distance $O M=h$ from

![](https://cdn.mathpix.com/cropped/2024_05_22_8a069bcd8684ea7ea3dbg-197.jpg?height=303&width=423&top_left_y=678&top_left_x=227)

Fig. 98

![](https://cdn.mathpix.com/cropped/2024_05_22_8a069bcd8684ea7ea3dbg-197.jpg?height=363&width=325&top_left_y=620&top_left_x=737)

Fig. 99

the $x O z$ plane, is found as the area of a square with side

$$
\begin{gathered}
M P=M N=\sqrt{a^{2}-h^{2}} \\
S(h)=a^{2}-h^{2}, \quad 0 \leqslant h \leqslant a
\end{gathered}
$$

The entire desired volume, according to formula (*), can be expressed by the integral

$$
V=8 \int_{0}^{a}\left(a^{2}-h^{2}\right) d h=\left.8\left(a^{2} h-\frac{h^{3}}{3}\right)\right|_{0} ^{a}=\frac{16}{3} a^{3}
$$