Asphere of radius r is cut by a plane h units above the equator, where
h < r.
find the volume of solid (spherical segment) above the plane.

Consider the top half of a sphere centered at the origin with radius , which can be described by the equation



and consider a plane



with . Call the region between the two surfaces . The volume of is given by the triple integral



Converting to polar coordinates will help make this computation easier. Set



Now, the volume can be computed with the integral



You should get

First one is 1 second one is 3