Evaluating Triple Integrals with Spherical Coordinates

In the spherical coordinate system the counterpart of a rectangular box is a spherical wedge

formula for triple integration in spherical coordinates.

where E is a spherical wedge given by

Formula says that we convert a triple integral from rectangular coordinates to spherical coordinates by writing

This formula can be extended to include more general spherical regions such as

Usually, spherical coordinates are used in triple integrals when surfaces such as cones and spheres form the boundary of the region of integration.

EXAMPLE 3: Evaluate , where B is the unit ball:

Since the boundary of B  is a sphere, we use spherical coordinates:

In addition, spherical coordinates are appropriate because

Note It would have been extremely awkward to evaluate the integral in Example 3 without spherical coordinates. In rectangular coordinates the iterated integral would have been

EXAMPLE 4: Use spherical coordinates to find the volume of the solid that lies above the cone   and below the sphere . See Figure

Notice that the sphere passes through the origin and has center . We write the equation of the sphere in spherical coordinates as

The equation of the cone can be written as

This gives  . Therefore the description of the solid E in spherical coordinates is

The volume of E is