Given a solid sphere, drill a cylindrical hole of length through the center of the sphere. Prove that the volume of the resulting ring is , where .
Proof. Let denote the radius of the sphere, and denote the radius of the cylindrical hole. Then the volume of the ring is the volume of the solid of revolution formed by rotating the area between the functions
about the -axis. Since the length of the hole is , we know ; thus, . So, we have,