I have a question about which order to put limits of integration for polar curves. Say I wanted to find the region enclosed by the curve $r= 2\cos\theta$ from $5\pi/3$ to $\pi/6$ (not the other way around). How would I do this? thanks!
It seems that you mean the area that lies at angles in $[5\pi/3,2\pi]$ or in $[0,\pi/6]$. To get this area, either perform those two integrations separately, or note that adding $2\pi$ makes no difference for your purposes, so you can integrate over $[5\pi/3,13\pi/6]$ in one go.