Evaluate the iterated integral?

Question

S is the region inside the circle r=4cosθ and outside the circcle r=2

Answer 1

First, find the points where they cross.

4 cos θ = 2

cos θ = 2/4 = 1/2

θ = -π/3, π/3

Then the area of the outside part is:

       π/3
1/2 ∫ (4 cos θ)² dθ =
      -π/3

        π/3
16/2 ∫ cos² θ dθ =
       -π/3

8 [θ/2 +1/4 sin 2θ] from -π/3 to π/3 =

8 [π/6 + 1/4 sin 2π/3 - (-π/6 + 1/4 sin -2π/3)] =

8 [π/3 + √3/8] = 8π/3 + √3

The area of the inside part is:

       π/3
1/2 ∫ 2² dθ = 4/2 [θ] from -π/3 to π/3 = 2[π/3 - (-π/3)] = 4π/3
      -π/3

Then the area should be 8π/3 + √3 - 4π/3 = 4π/3 + √3

Unless I messed something up.

(For the integral of cos² θ, I used an integral table; you'd want to use integration by parts.)