Polar area formula ($A=\frac{1}{2}{\int }_{\alpha }^{\beta }(r_2^2-r_1^2)d\theta$)

Suppose $0\le r_1(\theta )\le r_2(\theta )$ and the region is described without overlap by $\alpha \le \theta \le \beta$ and $r_1(\theta )\le r\le r_2(\theta )$. Integrating the polar area element gives its area:For a region bounded by a single radial graph and two rays, take $r_1=0$. If the two bounding graphs cross, split the angular interval at their intersections so that the larger radius is used consistently.