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390 Chapter 7 Applications of Definite Integrals<br />
7.2<br />
What you’ll learn about<br />
• Area Between Curves<br />
• Area Enclosed by Intersecting<br />
Curves<br />
• Boundaries with Changing<br />
Functions<br />
• Integrating with Respect to y<br />
• Saving Time with Geometric Formulas<br />
. . . and why<br />
The techniques of this section<br />
allow us to compute areas of<br />
complex regions of the plane.<br />
Areas in the Plane<br />
Area Between Curves<br />
We know how to find the area of a region between a curve and the x-axis but many times<br />
we want to know the area of a region that is bounded above by one curve, y f x, and<br />
below by another, y gx (Figure 7.3).<br />
We find the area as an integral by applying the first two steps of the modeling strategy<br />
developed in Section 7.1.<br />
1. We partition the region into vertical strips of equal width Δx and approximate each<br />
strip with a rectangle with base parallel to a, b (Figure 7.4). Each rectangle has area<br />
f c k gc k Δx<br />
for some c k in its respective subinterval (Figure 7.5). This expression will be nonnegative<br />
even if the region lies below the x-axis. We approximate the area of the region<br />
with the Riemann sum<br />
f c k gc k Δx.<br />
y<br />
Upper curve<br />
y f(x)<br />
y<br />
y f(x)<br />
y<br />
(c k , f (c k ))<br />
f (c k ) g(c k )<br />
a<br />
b<br />
x<br />
a<br />
b<br />
x<br />
a<br />
b<br />
x<br />
c k<br />
x<br />
Lower curve<br />
y g(x)<br />
y g(x)<br />
(c k , g(c k ))<br />
Figure 7.3 The region between y f (x)<br />
and y g(x) and the lines x a and<br />
x b.<br />
Figure 7.4 We approximate the region<br />
with rectangles perpendicular to the x-axis.<br />
2. The limit of these sums as Δx→0 is<br />
b<br />
f x gx dx.<br />
a<br />
Figure 7.5 The area of a typical rectangle<br />
is f (c k ) g(c k ) Δx.<br />
This approach to finding area captures the properties of area, so it can serve as a<br />
definition.<br />
DEFINITION<br />
Area Between Curves<br />
If f and g are continuous with f x gx throughout a, b, then the area between<br />
the curves y f (x) and y g(x) from a to b is the integral of f g from a to b,<br />
A b<br />
a<br />
f x gx dx.