Calculus- Early Transcendentals, 2021a

306 Applications of Integration
Guidelines for Area Between Two Curves
1. Find the intersection points.
2. Draw a sketch of the two curves.
3. Using the sketch determine which curve is the top curve and which curve is the bottom curve.
You may need to split the area up into multiple regions if the curves intersect multiple times
in [a,b].
4. Put the above information into the appropriate formula (once for each region):
Area =
∫ b
a
(top curve) − (bottom curve)dx, a ≤ x ≤ b.
5. Evaluate the integral using the Fundamental Theorem of Calculus (you should get a positive
number representing an area).
Example 8.4: Area Between Two Curves
Determine the area enclosed by y = x 2 , y = √ x, x = 0 and x = 2.
Solution. The points of intersection of y = x 2 and y = √ x are
x 2 = √ x → x 4 = x → x 4 − x = 0 → x(x 3 − 1)=0.
Thus, either x = 0orx = 1. Sketching the curves gives:
The area we want to compute is the shaded region. Since the top curve changes at x = 1, we need to
use the formula twice. For A 1 we have a = 0, b = 1, the top curve is y = √ x and the bottom curve is y = x 2 .
For A 2 we have a = 1, b = 2, the top curve is y = x 2 and the bottom curve is y = √ x.
For the first integral we have:
Area = A 1 + A 2 =
∫ 1
0
∫ 1
0
( √ x − x 2 )dx =
( √ x − x 2 )dx+
∫ 2
1
( 2
3 x3/2 − 1 3 x3 )∣ ∣∣∣
1
0
(x 2 − √ x)dx
= 1 3