Using polar coordinates to find the area of the region inside circle outside the cardioid

Question

Using polar coordinates to find the area of the region inside the circle $r = 1$ and outside the cardioid $r = 1 + \cos(\theta).$

Would the limits for $\theta$ be from -$\pi/2$ to $\pi/2$ and the limits for $r$ be from $1+\cos(\theta)$ to $1$?

If not what should they be?

Answer 1

The area is $$2\cdot \frac{1}{2} \int_{\frac{\pi }{2}}^{\pi } \left(1-(\cos \theta+1)^2\right) \, d\theta=2-\frac{\pi}{4}$$