Find the area of the region that is inside the curve r = sin θ and outside the curve r = 1 + cos θ. Your work must include the integral, but you may use your calculator to find...

Question

Find the area of the region that is inside the curve r = sin θ and outside the curve r = 1 + cos θ.

Your work must include the integral, but you may use your calculator to find the area to 3 decimal places. 

Answer 1

You can use a graphing utility to see that the curves intersect when Θ = π/2 and when Θ = π, so those become the bounds of integration. You are subtracting the area of the cardioid (r = 1 + cosΘ) in QII from the area of the circle in QII. You will use the formula for polar area A = ∫_a^b 1/2r²dΘ. So ...

A = ∫_π/2^π 1/2 [(sinΘ)² - (1 + cosΘ)²]dΘ ~ .4292