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

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