Find the area of the region by double integration

The question states:

Find the area of the region by double integration.

The region inside the circle $\displaystyle r=4cos(\Theta)$ but outside the circle $\displaystyle r=2$.

I set up the problem with $\displaystyle \int_{0}^{2\pi} \int_{2}^{4cos\Theta} rdrd\Theta$

However, the answer I got in the end was $\displaystyle 4\pi$, I'm assuming it's because of the way I set it up rather than a calculation error since I did it several times. The correct answer according to the textbook is $\displaystyle 4\pi/3 +2\sqrt{3}$.

Thanks, any help would be appreciated.