You can navigate through this E-Book using the menu to the left. For E-Books that have a Chapter/Section organization each option in the menu to the left indicates a chapter and will open a menu showing the sections in that chapter. Alternatively, you can navigate to the next/previous section or chapter by clicking the links in the boxes at the very top and bottom of the material.
Also, depending upon the E-Book, it will be possible to download the complete E-Book, the chapter containing the current section and/or the current section. You can do this be clicking on the E-Book, Chapter, and/or the Section link provided below.
For those pages with mathematics on them you can, in most cases, enlarge the mathematics portion by clicking on the equation. Click the enlarged version to hide it.
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.
|
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,
|