91Ó°ÊÓ

Show that the volume of a right square pyramid of height \(h\) and side length \(a\) is \(v=\frac{h a^{2}}{3}\) by using triple integrals.

Show that the volume of a regular right hexagonal prism of edge length \(\boldsymbol{a}\) is \(\frac{3 a^{3} \sqrt{3}}{2}\) by using triple integrals.

For the following two exercises, consider a spherical ring, which is a sphere with a cylindrical hole cut so that the axis of the cylinder passes through the center of the sphere (see the following figure). If the sphere has radius 4 and the cylinder has radius 2, find the volume of the spherical ring.

For the following problems, find the specified area or volume.The volume of the intersection between two spheres of radius 1 , the top whose center is \((0,0,0.25)\) and the bottom, which is centered at \((0,0,0)\).