Find the area of the region R inside the circle $ r=2 \cos \theta$ and outside the cardioid $r=2− 2 \cos \theta$ .

Question

I got $\theta\pi/3$ and $5\pi/3 $

and then the area I got was $-4\sqrt3-(8\pi)/3$

The area is not right, I used the area equation that takes integral of $1/2(f(\theta)^2-g(\theta)^2)$ from $\pi/3$ to $5\pi/3$

Answer 1

You integrated over the wrong interval. Note that if $r=2\cos\theta$ then $r$ is negative if $\pi/2\lt \theta\lt 3\pi/2$.

I would suggest using the symmetry, integrating over the top half, and doubling the result.

A careful diagram will help. You will end up integrating from $0$ to $\pi/3$. And it need not really be the integral of a difference, since the area of the relevant part of the circle can be written down without integrating.