In this section we want do take a look at triple integrals
done completely in Cylindrical Coordinates.
Recall that cylindrical coordinates
are really nothing more than an extension of polar coordinates into three
dimensions. The following are the
conversion formulas for cylindrical coordinates.
In order to do the integral in cylindrical coordinates we
will need to know what dV will become
in terms of cylindrical coordinates. We
will be able to show in the Change of Variables
section of this chapter that,
The region, E,
over which we are integrating becomes,
Note that we’ve only given this for E’s in which D is in the xy-plane. We can modify this accordingly if D is in the yz-plane or the xz-plane
as needed.
In terms of cylindrical coordinates a triple integral is,
Don’t forget to add in the r and make sure that all the x’s
and y’s also get converted over into
cylindrical coordinates.
Let’s see an example.
Just as we did with double integral involving polar
coordinates we can start with an iterated integral in terms of x, y,
and z and convert it to cylindrical
coordinates.
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Example 2 Convert
 into an integral in cylindrical coordinates.
Solution
Here are the ranges of the variables from this iterated
integral.
The first two inequalities define the region D and since the upper and lower bounds
for the x’s are and we know that we’ve got at least part of the
right half a circle of radius 1 centered at the origin. Since the range of y’s is we know that we have the complete
right half of the disk of radius 1 centered at the origin. So, the ranges for D in cylindrical coordinates are,
All that’s left to do now is to convert the limits of the z range, but that’s not too bad.
On a side note notice that the lower bound here is an
elliptic paraboloid and the upper bound is a cone. Therefore E is a portion of the region between these two surfaces.
The integral is,
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