Finding Area Between Two Polar Curves

[Lancelot59]

This particular problem is just confusing me in the setup. I need to find the area that is inside both:
r=sqrt(3)cos(theta) and r=sin(theta)

It makes a petal type shape. I was beating my head around for a while, but I reasoned that since the equation used to find the area cuts out in a straight line. I could just move in either direction following the appropriate functions and get the area by adding the two parts together:

$$\frac{1}{2}[\int_{\frac{\pi}{3}}^{\frac{\pi}{2}} (\sqrt{3}cos(\theta))^{2}) d\theta + \int_{0}^{\frac{\pi}{3}} (sin(\theta)^{2} d\theta]$$

It makes sense to me, but my final answer was 17pi\4 + 3sqrt(3)/8, however the book states the answer is 5pi/24 - sqrt3/4. Is my setup wrong, or did I just mess up with the integration?

 

[vela]

You just messed up the integration.

It's probably just a typo, but the book's answer should be $5\pi/24-\sqrt{3}/4$.

 

[Lancelot59]

It was a typo. The book had that answer. Thanks, I'll check my work again.