Calculus- Early Transcendentals, 2021a

304 Applications of Integration
Suppose we would like to find the area below f (x) =−x 2 + 4x + 3 and above g(x) =−x 3 + 7x 2 −
10x + 5 over the interval 1 ≤ x ≤ 2. We can approximate the area between two curves by dividing the area
into thin sections and approximating the area of each section by a rectangle, as indicated in figure 8.1. The
area of a typical rectangle is Δx( f (x i ) − g(x i )), so the total area is approximately
n−1
∑
i=0
( f (x i ) − g(x i ))Δx.
This is exactly the sort of sum that turns into an integral in the limit, namely the integral
∫ 2
f (x) − g(x)dx.
Then ∫ 2
f (x) − g(x)dx =
1
10
∫ 2
1
1
(−x 2 + 4x + 3) − (−x 3 + 7x 2 − 10x + 5)dx = 49
12 .
.
5
.
0
0 1 2 3
Figure 8.1: Approximating area between curves with rectangles.
This procedure can informally be thought of as follows.
Area Between Two Curves
Area =
∫ b
a
(top curve) − (bottom curve)dx, a ≤ x ≤ b.