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MI.8 Triple Integrals in Cylindrical and Spherical Coordinates

We have seen that in some cases double integrals are easier to solve using polar coordinates. In this section we see that some triple integrals are easier to evaluate using cylindrical or spherical coordinates.

Cylindrical Coordinates

Recall that cylindrical coordinates of a point P are $(r,\theta ,z)$ . r, $\theta ,$ and z are shown below:

figure 1 on 1052

Suppose that E is a type 1 region whose projection D on the xy-plane is conveniently described in polar coordinates. In particular, suppose that is continuous and:

This volume could look like the following:

figure 2 on 1052

D is given in polar coordinates by:

We know that type 1 regions are set up as follows:

But we also know how to evaluate double integrals in polar coordinates. So we just expand on this idea and use it in triple integrals as follows:

The formula above is called the formula for triple integration in cylindrical coordinates. It says that we convert a triple integral from rectangular to cylindrical coordinates by writing , and leaving z as it is, using the appropriate limits of integration and by replacing with $rdrd\theta dz.$ It is worthwhile to use this formula when E is a solid region easily described in cylindrical coordinates, and especially when the function involves the expression $x^{2}+y^{2}.$

Example

Evaluate

This iterated integral is a triple integral over the solid region

The graph of this region is shown below:

Picture of that region

The projection of E onto the xy-plane is the disk $x^{2}+y^{2}\leq 4.$ The lower surface of E is the cone and its upper surface is the plane This region has a much simpler description in cylindrical coordinates:

Therefore we have:

As shown above, the triple integral in cylindrical coordinates is much easier to solve.

Spherical Coordinates

Spherical coordinates take on the following relationship with rectangular coordinates:

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

Example of E

Just as with every other integral, we divide this area up into smaller and smaller pieces, and then we sum up these pieces. Next we take the limit of these sums to come up with the definition of the triple integral in spherical coordinates. Just as with polar coordinates and cylindrical coordinates we have to add something to the equation when we convert from rectangular to spherical, but instead of adding r we add

Where E is a spherical wedge given by

This formula can be extended into a more general formula

Example of a general region.

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