11. Two cylinders with equal radii of their bases are positioned so that their axes intersect each other at a right angle. Draw the figure resulting from the intersection of these cylinders. Calculate its volume if the radii of the cylinders are equal to $k$.
11. Let $O$ be the point of intersection of the axes of the cylinders (Fig. 108). A section plane, parallel to the axes and passing at a distance $x$ from point $O$, intersects the cylinders to form strips of width $2 \sqrt{R^{2}-x^{2}}$, and in the intersection with the considered body, it forms a square. The area of this section is $S(x)=4 R^{2}-4 x^{2}$. Since this area is a quadratic function of $x$, the volume can be calculated using Simpson's formula (see problem 39 below): \( V=\frac{h}{6}\left(S_{0}+4 S_{m}+S_{n}\right) \). Here, \( S_{0}=S_{n}=0 \), \( S_{m}=4 R^{2} \), \( h=2 R \), so \( V=\frac{2 R}{6} \cdot 16 R^{2} ; V=\frac{16}{3} R^{3} \). The answer can also be obtained by directly integrating the function of the section area \( S(x) \).