Determine or find out the area of the inner loop of r = 2 + 4 cosθ.

Solution

We can graphed this function back while we first started looking at polar coordinates.  For this example we'll as well need to know the values of θ in which the curve undergoes the origin.  We can obtain these by setting the equation equal to zero and solving.

0 = 2 + 4cosθ

Cosθ = -1/2                      => θ = 2Π/3, 4Π/3 

Here is the diagram of this curve along with the inner loop shaded in.

Can you observe why we required to know the values of θ in which the curve goes through the origin? 

These points describe where the inner loop starts and ends and therefore are also the limits of integration in the formula.  

Thus, the area is then,