Calculus- Early Transcendentals, 2021a

8.2. Area Between Curves 307
Thus,
Area = 1 3 + ( 1
3 x3 − 2 3 x3/2 )∣ ∣∣∣
2
1
[( )
= 1 3 + 8
3 − 2(√ 2) 3 ( 1
−
3 3 − 2 ) ] = 10 − 4√ 2
3
3
♣
Example 8.5: Area Between Sine and Cosine
Determine the area enclosed by y = sinx and y = cosx on the interval [0, 2π].
Solution. The curves y = sinx and y = cosx intersect when:
sinx = cosx → tanx = 1 → x = π + πk, k an integer.
4
We have the following sketch:
The area we want to compute is the shaded region. The top curve changes at x = π/4 andx = 5π/4,
thus, we need to split the area up into three regions: from 0 to π/4; from π/4 to5π/4; and from 5π/4 to
2π.
∫ π ∫ 5π ∫
4
4
2π
Area = (cosx − sinx)dx+ (sinx − cosx)dx+ (cosx − sinx)dx
π 5π
0
4
4
π/4
5π/4
2π
= (sinx + cosx) ∣ +(−cosx − sinx) ∣ +(sinx + cosx) ∣
=
(√
2 − 1
)
+
= 4 √ 2
∣
0
(√ √ )
2 + 2 +
(
1 + √ 2
Sometimes the given curves are not functions of x. In this instances, it may be more useful to use the
following.
∣
)
π/4
∣
5π/4
♣