Polar Change of Variables

System

• $x=r\cos \theta$
• $y=r\sin \theta$

Usage

$$\int {\int }_{D}f(x,y)dA=\int {\int }_{D^{\prime }}f(r\cos \theta ,r\sin \theta )rd{A}^{\prime }$$

General Double Change of Variables where $J=r$

Use when

• $D$ is bounded by circles, rays, or polar graphs
• $f$ depends on $\sqrt{x^2+y^2}=r$ or $y/x=\tan \theta$

Natural Range

• $r\in [0,\infty )$
• $\theta \in [0,2\pi ]$