Chapter 14: Multiple Integrals
Summary:
The previous chapter focused upon taking derivatives and limits of
multivariable functions. In this chapter, the integration of multivariable functions
is considered. This naturally leads to integrating functions with respect to two or
three variables. The integration of these types of functions over their domains is
described best using double and triple integrals. Double integrals are useful for
calculating the area of a two dimensional region or its center of mass. They are
also useful for calculating the surface area of a surface determined by a function
z
=
f
(
x
,
y
)
.
Triple integrals allow the mass of an object to be determined and
consequently the location of its center of mass.
In many problems, situations in two and three dimensions are not easily described
using Cartesian or rectangular coordinates. In some cases, problems may be more
easily considered if the coordinates are given as polar coordinates in two dimen
sions or by cylindrical coordinates or spherical coordinates in three dimensions.
Just as
u
substitution played an important role in the development of integration in
one dimension, using a change of variables is important in both double and triple
integral problems. In particular, when switching from rectangular coordinates to
either polar, cylindrical or spherical coordinates, a change of variables is being
performed to switch from one set of coordinates to another.
OBJECTIVES: After reading and working through this chapter
you should be able to do the following:
1. Set up double integrals and evaluate double integrals (§14
.
1).
2. Write double integrals with nonconstant limits of integration (§14
.
2).
3. Set up double integrals over nonrectangular regions (using rectangular, §14
.
2,
or polar coordinates, §14
.
3).
4. Find the volume under a surface using double integrals (§14
.
1
−
14
.
3).
5. Switch the order of integration in double integrals (§14
.
2
,
14
.
3).
6. Change between rectangular and polar coordinates in double integrals (§14
.
3).
7. Calculate area using double integrals (§14
.
2
,
14
.
3).
8. Define surfaces parametrically (§14
.
4).
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9. Express a parametric surface in vector form (§14
.
4).
10. Find tangent planes to parametric surfaces and surface area of a parametric
surface (§14
.
4).
11. Calculate volume under a surface using a triple integral (§14
.
5).
12. Determine appropriate limits of integration for triple integrals (§14
.
5).
13. Set up and evaluate triple integrals using both cylindrical and spherical co
ordinates (§14
.
6).
14. Convert between different coordinate systems for triple integrals (§14
.
6).
15. Find the Jacobian and perform general changes of variables in double and
triple integrals (§14
.
7).
16. Find the centroid of a lamina using double integrals (§14
.
8).
17. Find the centroid of a solid using triple integrals (§14
.
8).
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 Spring '10
 lee
 dy dx, dx dy, Jacobian

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