Triple Integrals in Cylindrical and Spherical Coordinates

Triple integrals in cylindrical coordinates Suppose we want to compute the triple integral

, where

is a simple xy-solid over a general polar region

. It is more convenient to change the triple integral into cylindrical coordinates as follows:

where

is bounded from below and above by

and

respectively over the polar region

. Example: Find the volume of the cylinder

between the planes

and

. Answer: Let

be the part of cylinder

between

and

. Using cylindrical coordinates, the triple integral that equals the volume of

can be expressed as follows:

(Note: The cylinder has radius 3.)

The applet below shows that the solid

, which is the part of the cylinder between two planes.

Triple integrals in spherical coordinates Sometimes we might want to express a triple integral in spherical coordinates. Suppose

is defined on a "spherical box"

. We then divide each interval into

subdivisions such that

. Label the spherical subbox from

. Let

be the centre point in the

spherical subbox. The volume

of the

spherical subbox equals

, as shown in the diagram below. The triple integral in spherical coordinates is the limit of the following Riemann sum:

Therefore, we can rewrite the triple integral as an iterated integral in spherical coordinates as follows:

Example: Find the volume of the solid

bounded from above and below by the sphere

and the cone

. Answer: We first express the sphere and cone in spherical coordinates:

and

respectively. Therefore,

. Therefore, we have the following:

The applet below shows the solid

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