Calculus- Early Transcendentals, 2021a

11.3. Areas in Polar Coordinates 409
This first example makes the process appear more straightforward than it is. Since points have many
different representations in polar coordinates, it is not always so easy to identify points of intersection.
Example 11.12: Shaded Area
Find the shaded area in the first graph of figure 11.9 as the difference of the other two shaded areas.
The cardioid is r = 1 + sinθ and the circle is r = 3sinθ.
Solution. We attempt to find points of intersection:
1 + sinθ = 3sinθ
1 = 2sinθ
1/2 = sinθ.
This has solutions θ = π/6 and5π/6; π/6 corresponds to the intersection in the first quadrant that we
need. Note that no solution of this equation corresponds to the intersection point at the origin, but fortunately
that one is obvious. The cardioid goes through the origin when θ = −π/2; the circle goes through
the origin at multiples of π, starting with 0.
Now the larger region has area
∫
1 π/6
(1 + sinθ) 2 dθ = π 2 −π/2
2 − 9 √
3,
16
and the smaller has area
∫
1 π/6
(3sinθ) 2 dθ = 3π 2 0
8 − 16√ 9 3,
so the difference is the area we seek, which is π/8.
♣
Figure 11.9: An area between curves.
Exercises for 11.3
Exercise 11.3.1 Find the area enclosed by the curve.