Find the area of the region enclosed by the inner loop of the curve.

Question

r = 4 + 8 sin(θ)

Answer 1

r=0 at θ=7π/6, 11π/6
So, using symmetry, the area is

a = 2∫[7π/6,3π/2] 1/2 (4+8sinθ)^2 dθ = 8(2π-3√3) ≈ 8.6963