Originally Posted by

**greenstupor** I'm currently reading an AP Calculus BC review book, and one of its example problems has to do with finding the area of the inner loop of a limaçon.

The formula for the limaçon is $\displaystyle r=1+2\cos\theta$. I'm supposed to find the area of just its inner loop. The book begins its problem walkthrough by stating that "in the inner loop, $\displaystyle r$ sweeps out the region as $\displaystyle \theta$ goes from $\displaystyle \frac{2\pi}{3}$ to $\displaystyle \frac{4\pi}{3}$." Those would, therefore, be the limits of integration.

I'm really shaky on polar coordinate geometry, though, so I have no idea how you're supposed to have figured that out. How do you find the $\displaystyle \theta$ values on a polar coordinate graph? I'm familiar with the basic polar-cartesian conversion equations $\displaystyle x=r\cos\theta$, $\displaystyle y=r\sin\theta$, $\displaystyle r=\sqrt{x^2+y^2}$, and $\displaystyle \theta=\arctan{\frac{y}{x}}$, but I don't know how to find $\displaystyle \theta$ values when the polar graph is as complicated as a looped limaçon.

Thanks for any help you could give me!