How to find the area inside the larger loop and outside the smaller loop of the limacon $r=\frac{1}{2} +\cos \theta$?

Question

How to find the area inside the larger loop and outside the smaller loopof the limacon $r=\frac{1}{2} +\cos \theta$?

Once the integrals are set up, I know how to solve them, but I'm having difficulty setting the integrals up.

Answer 1

The area of one half of the outer loop is given by $\displaystyle\frac{1}{2} \int \limits_{0}^{2\pi/3} (\frac{1}{2} +\cos \theta)^2 d\theta$. See This.

The area of one half of the inside loop is given by $\displaystyle\frac{1}{2} \int \limits_{\pi}^{4\pi/3} (\frac{1}{2} +\cos \theta)^2 d\theta$. See This.

Now, we subtract the outer loop from the inner loop and multiply by $2$ to account for symmetry:

$$2\left(\displaystyle\frac{1}{2} \int \limits_{0}^{2\pi/3} (\frac{1}{2} +\cos \theta)^2 d\theta-\displaystyle\frac{1}{2} \int \limits_{\pi}^{4\pi/3} (\frac{1}{2} +\cos \theta)^2 d\theta\right)$$