Finding the Area Between Two Curves Written in Polar Coordinates
Problems of this kind are variable and each one may demand a different approach.
In the diagram belowis the curveandis the curveW e have to find the shaded are and this must be done in several stages.
We find the points A and B of intersection of the two curves, specifically, we find thevalues andHaving found these, the shaded area will consist of the sum of two integrals:
We integrate forthe cardioid betweenandand forbetweenand Notice thatis negative.
The pints of intersections are the solutions to
Forwe integrate betweenandor anticlockwise from
The area of the circle between A and B is
The shaded area is the sum of these