Polar area help

Question

Find the area that it encloses.

(a) r = 1 − sin 𝜃

Find the area that it encloses.

(b) r = 7 + 2 cos 𝜃

Answer 1

I'll try to make up for my last answer:

1) the graph is a cardioid with the two lobes up. The full curve has the domain 0 ≤ θ ≤ 2π

The integral dA in polar coordinates is the double integral of rdrdθ and if we integrate r first, we obtain

the integral from 0 to 2π of 1/2r²dθ or integral from 0 to 2π of 1/2(1-sinθ)²dθ

sin²θ = 1/2(1-cos(2θ) so the integrand becomes 1/2(1 - 2sinθ +1/2(1-cos(2θ)) or 3/4 - sinθ -1/4cos(2θ)

This integral is (3/4)θ + cosθ -(1/8)sin2θ evaluate from 0 to 2π = 3π/2

2) r = 7 + 2cosθ is part of the family of curves, but this one looks like a lopsided circle to the right. The technique is the same, so I leave it to you.

Answer 2

(a) s = 2·0.5∫_-π/2^π/2(1 - sinθ)²dθ = ∫_-π/2^π/2(1 - 2sinθ + sin²θ)dθ = ∫_-π/2^π/2(3/2 - 2sinθ - cos2θ/2)dθ =

(3/2θ + 2cosθ - sin2θ/4)_-π/2^π/2 = 3/2π

(b) s = 2·0.5∫₀^π(7 + 2cosθ)²dθ = ∫₀^π(49 + 28cosθ + 4cos²θ)dθ = ∫₀^π(51 + 28cosθ + 2cos2θ)dθ =

(51θ + 28sinθ + sin2θ)₀^π = 51π