# 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.

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watko@athena.mit.edu