Can someone find the area between r =1 and r = 1 - sin (theta) (Inside the cartoid, outside the circle) ?

Question

I set up an integral and got the area to be 4 + pi/2 . Is this correct?

(This was on a math test, so I don't have my work in front of me)

Answer 1

First we need to find the interval over which to integrate.

r = 1 - sinθ = 1
sinθ = 0
θ = π, 2π

We need to integrate over the interval [π,2π]. These are the angles where the radius of the cardioid is greater than that of the circle.

Area = ∫(1/2)(R² - r²) dθ = (1/2)∫[(1 - sinθ)² - 1²] dθ

= (1/2)∫[1 - 2sinθ + sin²θ - 1] dθ

= (1/2)∫[sin²θ - 2sinθ] dθ = (1/2)∫[(1 - cos2θ)/2 - 2sinθ] dθ

= ∫[1/4 - (cos2θ)/4 - sinθ] dθ

= θ/4 - (sin2θ)/8 + cosθ | [Evaluated from π to 2π]

= [2π/4 - 0 + 1] - [π/4 - 0 - 1] = 2π/4 + 1 - π/4 + 1

= π/4 + 2