5128_Ch07_pp378-433

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