29.30. Find the volume of the figure formed by the intersection of two straight circular cylinders of radius $R$, the axes of which are perpendicular and intersect.
29.30. Answer: $\frac{16}{3} R^{3}$. Let's introduce a coordinate system, directing the axes $O x$ and $O y$ along the axes of the cylinders. The intersection of the cylinders is described by the inequalities $x^{2}+z^{2} \leqslant R$ and $y^{2}+z^{2} \leqslant R$. Its section by the plane $z=z_{0}$ is described by the inequalities $x_{0} \leqslant R^{2}-z_{0}^{2}$ and $y_{0} \leqslant R^{2}-z_{0}^{2}$. Thus, the section is a square with side $2 \sqrt{R^{2}-z_{0}^{2}}$. Its area is $4\left(R^{2}-z_{0}^{2}\right)$. Therefore, the volume of the considered figure is

$$
\int_{-R}^{R} 4\left(R^{2}-z^{2}\right) d z=8 R^{3}-\left.4 \frac{z^{3}}{3}\right|_{-R} ^{R}=8 R^{3}-\frac{8}{3} R^{3}=\frac{16}{3} R^{3}
$$