Multivariate Calculus - Bruce E. Shapiro

136 LECTURE 17. DOUBLE INTEGRALS OVER GENERAL REGIONS
Theorem 17.2 Additivity. Suppose that f(x, y) is integrable on S, that S =
A ∪ B, and that any overlap between A and B occurs only on a smooth curve. Then
f(x, y)dA = f(x, y)dA + f(x, y)dA
S
A
B
Figure 17.2: The integral is additive over regions that share (at most) a common
smooth curve. Here ∬ S f = ∬ A f + ∬ B f.
Theorem 17.3 Comparison Property. Suppose that f(x, y) and g(x, y) are integrable
functions on a set S such that
for all (x, y) ∈ S. Then
f(x, y) ≤ g(x, y)
f(x, y)dA ≤ g(x, y)dA
S
The simplest non-rectangular regions to integrate over are called x -simple and y-
simple regions, as illustrated in the following example. A region is called x-simple
if it can be expressed as a union of line segments parallel to the x-axis; it is called
y-simple if it can be expressed as a union of line segments parallel to the y-axis.
The following example considers one y-simple set.
Example 17.1 Set up the integral
s
S
f(x, y)dA
where S is the set bounded above by the curve y = x 2 , below by the line y = −3x,
on the left by the line x = 2, and on the right by the line x=10.
Solution. As x increases from x=2 to x=10, we can trace out the figure with
a vertical rectangle that goes from the lower boundary y = −3x to the upper
boundary y = x 2 . In other words: For all x between 2 and 10 (this is the outer
integral), include all y between y = −3x and y = x 2 . Therefore
s f(x, y)dA = ∫ 10
2
∫ x 2
−3x
f(x, y)dx
Revised December 6, 2006. Math 250, Fall 2006