Area between two polar curves

 Goal: find the area between 2 polar curves

Area between 2 polar curves

The procedure to find the area  between 2 polar curves is similar to that of the area of 2 curves in the system of coordinates in the cartesian plane, We find the points of intersection between the 2 curves and identify the functions that define the outer curve and the inner curve respectively.

Example

Find the area outside of the cardioid r = 2 + 2 sinθ and inside the circle r = 6 sinθ

Solution

 First, draw a graph containing both curves

To find the limits of integration, let's find the points of intersection by setting the 2 functions equal to each other and solving for θ

The solutions of this equation are θ = ℼ/6 and θ = 5ℼ/6, which are the limits of integration. The graph of the circle, in red.. is the outer curve. The graph of the cardioid, in blue, is the inner curve. To find the area between the 2 curves, let's subtract the area of the cardioid from that of the circle.

Practice

Find the area inside the circle r = 4 cosθ and outside the circle r = 2.