Find the area of the shaded region

Question

Write down the name of the curves and shade the region that is outside of r = 4 and inside of

r = 4 - 2 sinθ. Hence, find the area of the region shaded.

Answer 1

r = a - b sin θ

a / b =2 => The name of curve is convex limaçon

Area

= region that is inside of r = 4 - 2 sinθ - region that is inside of r = 4 (only when θ ∈ [π,2π] )

= 1/2 ∫ (from π to 2π) (4 - 2 sin θ)² d θ - 1/2 ∫ (from π to 2π) 4² d θ

= 1/2 ∫ (from π to 2π) (16 - 16 sin θ + 4 sin²θ) d θ - 8 π

= 1/2 [ (16 θ + 16 cos θ ](from π to 2π) + 1/2 ∫ (from π to 2π) (4 sin²θ) d θ- 8 π

= 1/2 [ (16 π + 32 ] + 1/2 ∫ (from π to 2π) (4 sin²θ) d θ - 8 π

= 1/2 [ (16 π + 32 ] + ∫ (from π to 2π) (1- cos 2θ) d θ - 8 π (Using sin²x = (1 - cos 2x) / 2 )

= 16 + 1/2 ∫ (from 2π to 4π) (1- cos ϕ) d ϕ (Let ϕ = 2θ)

= 16 + 1/2 (ϕ - sin ϕ)] ((from 2π to 4π))

= 16 + 1/2 (2π - 0)

= 16 + π (square units)