Objectives:

1.
Be comfortable setting up and computing triple integrals in cylindrical and spherical coordinates.
2.
Understand the scaling factors for triple integrals in cylindrical and spherical coordinates, as well as where they come from.
3.
Be comfortable picking between cylindrical and spherical coordinates.

Cylindrical Coordinates

Recap Video

Here is a video highlights the main points of the section.

To summarize:

Example Video

Here is an example of setting up the bounds for a triple integral in cylindrical coordinates.

Problems

Evaluate , where is the region , .
Let be the region given by , . Evaluate .

Spherical Coordinates

Throughout this section, you will need to use “” to denote “.”

Recap Video

Here is a video highlights the main points of the section.

To summarize:

Example Video

Here is an example of setting up the bounds for a triple integral in spherical coordinates.

Problems

NOTE: Use “p” instead of in your answer inputs. For example input for .

Let be the portion of the sphere with . Evaluate .
Let be the region and . Find the volume of .

Picking a Coordinate System

For the following problems, you should decide whether to use Cartesian, cylindrical, or spherical coordinates to evaluate the triple integral. It would be good practice for you to try all three and see which ones are realistic options for each problem.

You once did the next problem as a double integral. Now do it as a triple integral and convince yourself it is the same thing.

Let be the region bounded above by and below by . Then the volume of is:
Let be the portion of above the plane . Consider the integral .
  • In cylindrical coordinates, the integral would be
  • In spherical, the integral would be (input “p” for phi):
  • Evaluating either way gives an answer of: .
Let be the region bounded by , , , , and . Then .
Let be the region in the first octant bounded by and . Then
Let be the solid given by and . The volume of is:
Let be the region given by and . Then .