In these two videos you are shown a couple of examples on how to find the area bounded by a polar curve. The two curves are a cardioid with polar equation r=a(1+cosθ) and one loop of the curve r=a cos 3θ

Area bounded by the cardioid r=a(1 + cos θ)

Polar Curves - Area bounded by a Cardioid : ExamSolutions Maths Revision - youtube Video

Area of a loop of the curve r=a cos 3θ

Area of a Loop in a Polar Curve : ExamSolutions Maths Revision - youtube Video