Polar coordinates: what is the area of the region inside the inner loop of $r = \cos (\theta) - \frac12$?

Question

I'm struggling plotting $r = \cos (\theta) - \frac12$. I've done it in Cartesian but I can't quite get in polar coordinates. I know it is supposed to be a loop but how do I get it? Being that I have to find the area of the inner loop, what will be end points of my integral?

Answer 1

Looking at the function, we can see that the inner loop starts and ends at the origin. To solve for $\theta$, set r = 0. The $\theta$ values we receive are $\frac{\pi}{3}$ and $\frac{5\pi}{3}$.