Eroxl's Notes
Evaluating Triple Integrals in Spherical Coordinates

Evaluating double integrals in spherical coordinates is useful when the region of integration is more naturally described in spherical coordinates such as spheres or objects with three lines of radial symmetry.

This method transforms the integral from Cartesian coordinates to cylindrical coordinates .

Transformation

The relationship between Cartesian and spherical coordinates is given by:

where , , and .

The area element in polar coordinates is . The additional factor of arrises from the Jacobian of the transformation from Cartesian to sphericla coordinates.