# Vector Calculus Independent Study Path

## Unit 6: Double and Triple Integrals

Single integrals, the integrals you learned all about in calculus,
find the area under the graph of a function of one variable.
Double integrals, the integrals you will learn about in this section,
find the volume under the graph of a function of two variables.
How do you calculate a double integral? You take two integrals
and call me in the morning. No, seriously, that's what you do.
And triple integrals? Well, they find the hyper-volume under the
graph of a function of three variables. I mean, duh.

In this unit, you will learn:

- The definition of double and triple integrals in terms of a limit
of Riemann sums.
- How to calculate double and triple integrals as iterated (nested)
integrals.
- How to change the order of an integration.
- How to change the variables of an integration. In particular,
you will learn how to do triple integrals in spherical and cylindrical
coordinates.
- How to apply double and triple integrals to calculate masses,
centers of mass, average values, and moments of inertia.

For more detailed instructions, see the Xdvi
or PDF pages.
## Suggested Procedure

- Read and do some problems from
- Rogers Chapters 14 - 18,
- Marsden and Tromba chapters 5 and 6, or
- Simmons, chapter 20

- Take the Sample Test, Xdvi or PDF.
- Take a unit test.

Vector Calculus Independent Study Path |
Vector Calculus Index |
World Web Math Main Page

watko@athena.mit.edu
Last modified November 5, 1998