Multiple integrals: Triple integrals
Change of variables in triple integrals: spherical and cylindrical coordinates
Triple integral in other coordinates Suppose , and is an invertible mapping from a bounded region in the -plane to a region in the plane and suppose the functions , and have continuous derivatives on . If the triple integral exists (i.e. is integrable on ) and if , then is integrable on and with the Jacobian defined as
Example of scaling Suppose you want to calculate the volume of an ellipsoid with . Under the transformation the unit sphere is mapped onto the ellipsoid . The Jacobian can easily be calculated: The volume of an ellipsoid can now be calculated as follows:
Integrals in cylindrical coordinates Cylindrical coordinates are actually polar coordinates and with an extra dimension . As with polar coordinates, and therefore apply
First note that the region of integration can be described in cylindrical coordinates with the region of integration . The requested triple integral can now be calculated as an iterated integral in the following way:
Integrals in spherical coordinates Spherical coordinates are defined as in the figure below, with the following formulas
In this case the Jacobian of the coordinate transformation is and therefore
First note that the region of integration can be described in spherical coordinates with the region of integration The requested triple integral can now be calculated as iterated integral in the following way: