Calculus- Early Transcendentals, 2021a

14.2. Double Integrals in Polar Coordinates 497
Example 14.6
Find the area outside the circle r = 2 and inside r = 4sinθ; see Figure 14.9.
Solution. The region is described by π/6 ≤ θ ≤ 5π/6and2≤ r ≤ 4sinθ, so the integral is
∫ 5π/6 ∫ 4sinθ
π/6
2
1rdrdθ =
=
∫ 5π/6
π/6
∫ 5π/6
π/6
= 4 3 π + 2√ 3.
∣
1 ∣∣∣
4sinθ
2 r2 dθ
2
8sin 2 θ − 2dθ
♣
Figure 14.9: Finding area by computing volume.
Exercises for 14.2
Exercise 14.2.1 Find the volume above the xy-plane, under the surface r 2 = 2z, and inside r = 2.
Exercise 14.2.2 Find the volume inside both r = 1 and r 2 + z 2 = 4.
Exercise 14.2.3 Find the volume below z = √ 1 − r 2 and above the top half of the cone z = r.
Exercise 14.2.4 Find the volume below z = r, above the xy-plane, and inside r = cosθ.
Exercise 14.2.5 Find the volume below z = r, above the xy-plane, and inside r = 1 + cosθ.
Exercise 14.2.6 Find the volume between x 2 + y 2 = z 2 and x 2 + y 2 = z.
Exercise 14.2.7 Find the area inside r = 1 + sinθ and outside r = 2sinθ.
Exercise 14.2.8 Find the area inside both r = 2sinθ and r = 2cosθ.
Exercise 14.2.9 Find the area inside the four-leaf rose r = cos(2θ) and outside r = 1/2.