Triple Integrals in Cylindrical Coordinates

Triple integrals in cylindrical coordinates occur frequently in applications.  The use of cylindrical coordinates is often due to expressing a triple integral as a double integral of an integral and then transforming the double integral in polar coordinates.  However, here we are going to use a more sophisticated approach--one that is analogous to the approach taken in the change of variable section.

To begin with, the transformation to cylindrical coordinates is a 3-dimensional coordinate transformation of the form

Thus, as was the case with the two-dimensional transformations and the area differential, the volume differential is determined by the determinant of the Jacobian of the transformation.

Let's use Maple to calculate the Jacobian determinant.  First, we define the transformation:

>	T:=<r*cos(theta),r*sin(theta),z>;

>

Now let's use the Jacobian command from the VectorCalculus  package to calculate the Jacobian determinant.  The Jacobian  command returns both the Jacobian matrix and the Jacobian determinant, so we assign 2 labels--J and Det_J--to the output from Jacobian .   We then simplify the determinant output.

>	(J,Det_J):=Jacobian(T, [r,theta,z], 'determinant' ); Det_J:=simplify(Det_J);

>

The result is that the volume differential in spherical coordinates is given by

Let's look at an example.

Example:

A cylindrical tank with a height of 20 meters and a radius of 5 meters contains a mixture of sand and gravel, so that its mass density is

In particular, the denser gravel had settled to the bottom of the tank to some extent.

(commands used to generate the image above)