area between two polar curves

Question

I am trying to find the area between the following two curves given by the following polar equations: $r=\sqrt{3}\cos\theta$ and $r=1+\sin\theta$.

I did the following: First, I found the points of intersection: The curves intersect each other at the origin and when $\theta=\pi/6$. Then the area between the two curves is: $\int_{3\pi/2}^{2\pi+\pi/6}\frac{1}{2}(1+\sin\theta)^{2}d\ d\theta+\int_{\pi/6}^{\pi/2}\frac{1}{2}(\sqrt{3}\cos\theta)^{2}\ d\theta$. I am not sure if this is right or not. I appreciate if anyone can tell me whether there is a mistake somewhere in my solution. Also, if you know others methods of doing the problem, please let me know

Answer 1

A satisfactory answer can be found in the comments, hence this CW answer.