The step by step process of the given.

Question

Answer must be in EXACT form only.

1. Determine the area common to r= 3+3 sin θ and r=3.

Answer 1

Limits of integration: 3 = 3 + 3sinθ; sinθ = 0, θ = 0, θ = π, θ = 2π;

Area A = 1/2∫₀^π3²dθ + 1/2∫_π^2π(3 + 3sinθ)²dθ = 1/2·9π + 1/2∫_π^2π(9 + 18sinθ + 9sin²θ)dθ = 9π/2 +

9/2∫_π^2π(3/2 + 2sinθ - 1/2cos2θ)dθ = 9π/2 + 9/2(3/2θ - 2cosθ - 1/4sin2θ)_π^2π = 9π/2 + 9/2(3π - 2 - 3π/2 - 2) =

9/2(π + 3π/2 - 4) = 9(5π - 8)/4