Area between polar curves

To get the area between the polar curve $r=f(\theta)$ and the polar curve $r=g(\theta)$, we just subtract the area inside the inner curve from the area inside the outer curve. If $f(\theta) \ge g(\theta)$, this means $$\frac{1}{2}\int_a^b f(\theta)^2 - g(\theta)^2 d\theta.$$ Note that this is NOT $\frac{1}{2}\int_a^b [f(\theta)-g(\theta)]^2 d\theta$!! You first square and then subtract, not the other way around.

As with most ``area between two curves'' problems, the tricky thing is figuring out the beginning and ending angles. This is typically where $f(\theta)=g(\theta)$.