Calculus- Early Transcendentals, 2021a

8.2. Area Between Curves 305
More formally, the area A of the region bounded by the curves y = f (x) and y = g(x) and the lines
x = a and x = b is:
A =
∫ b
a
| f (x) − g(x)| dx.
Example 8.3: Area between Curves
Find the area between f (x)=−x 2 + 4x and g(x)=x 2 − 6x + 5; the curves are shown in figure 8.2.
Solution. Here we are not given a specific interval, so it must be the case that there is a “natural” region
involved. Since the curves are both parabolas, the only reasonable interpretation is the region between the
two intersection points, which can be computed as:
5 ± √ 15
.
2
If we let a =(5 − √ 15)/2andb =(5 + √ 15)/2, the total area is
∫ b
a
∫ b
−x 2 + 4x − (x 2 − 6x + 5)dx = −2x 2 + 10x − 5dx
a
= − 2x3
b
3 + 5x2 − 5x
∣
a
= 5 √ 15.
after a bit of simplification.
♣
5
0
1 2 3 4 5
.
−5
Figure 8.2: Area bounded by two curves.
.
Some general guidelines to compute the area between two curves follows.