文档介绍：Contents
CHAPTER 14 Multiple Integrals
Double Integrals
Changing to Better Coordinates
Triple Integrals
Cylindrical and Spherical Coordinates
CHAPTER 15 Vector Calculus
Vector Fields
Line Integrals
Green's Theorem
Surface Integrals
The Divergence Theorem
Stokes' Theorem and the Curl of F
CHAPTER 16 Mathematics after Calculus
Linear Algebra
Differential Equations
Discrete Mathematics
Study Guide For Chapter 1
Answers to Odd-Numbered Problems
Index
Table of Integrals
CHAPTER 14
Multiple Integrals
Double Integrals 4
This chapter shows how to integrate functions of two or more variables. First, a
double integral is defined as the limit of sums. Second, we find a fast way pute
it. The key idea is to replace a double integral by two ordinary "single" integrals.
The double integral Sf f(x, y)dy dx starts with 1f(x, y)dy. For each fixed x we integ-
rate with respect to y. The answer depends on x. Now integrate again, this time with
respect to x. The limits of integration need care and attention! Frequently those limits
on y and x are the hardest part.
Why bother with sums and limits in the first place? Two reasons. There has to be
a definition and putation to fall back on, when the single integrals are difficult
or impossible. And also-this we emphasize-multiple integrals represent more than
area and volume. Those words and the pictures that go with them are the easiest to
understand. You can almost see the volume as a "sum of slices" or a "double sum of
thin sticks." The true applications are mostly to other things, but the central idea is
always the same: Add up small pieces and take limits.
We begin with the area of R and the volume of by double integrals.
A LIMIT OF SUMS
The graph of z =f(x, y) is a curved surface above the xy plane. At the point (x, y) in
the plane, the height of th