 Polar area element

For polar coordinates $x=r\cos \theta$, $y=r\sin \theta$ with $r\ge 0$, the Jacobian determinant is $r$. The change of variables formula therefore gives the area element $dA=rdrd\theta$. Integrating over the radial interval $r_1(\theta )\le r\le r_2(\theta )$ gives $[r_2(\theta {)}^2-r_1(\theta {)}^2]/2$ per unit angle. Bounds must describe the intended region without counting it more than once.