Area of a Cardioid

Question

Find the area of the region R lies inside the cardioid r=3(1+sinX) and outside the circle r=3sinX

Answer 1

Recall that the area inside a polar curve is given by A = ∫1/2 r² dθ

Since the circle lies entirely inside the cardioid, and the regions is symmetric, we can just double the area inside the first quadrant. So,

A = 2∫₀^π/2 1/2 ((3(1+sinx))² - (3sinx)²) dx = 18 + 9π/2