Confused on a calculus question please help

Question

Find the area of the region outside the circle r₁ and inside the limacon r₂. Round to two decimal places.

r₁ = 3



r₂ = 2+2 cos θ

Answer 1

We can solve algebraically for the bounds of integration, which are the theta values of the points of intersection between the two polar curves. These points lie in Quadrant IV and I:

2 + 2 cosΘ = 3

cosΘ = 1/2

Θ = - π/3 or π/3

For Θ in this range, r₂ > r₁, so we integrate 1/2(r₂ - r₁)² dΘ as follows:

A = ∫_-π/3^π/3 [1/2(2 + 2 cosΘ - 3)²] dΘ

= 1/2 ∫_-π/3^π/3 [(2 cosΘ - 1)²] dΘ

= 1/2 ∫_-π/3^π/3 [4 cos²Θ - 4cosΘ + 1] dΘ ...

Note: 4cos²Θ can be integrated by using the double-angle identity cos²Θ = (cos2Θ + 1) / 2