极坐标和直角坐标的积分转换

如下图所示，当$\theta 不变，\rho 增大为\rho +\Delta \rho (即{\rho }_1)时$, 面积的增量为：

$$\begin{gathered} \frac{1}{2}\theta (\rho +\Delta \rho {)}^2-\frac{1}{2}\theta {\rho }^2= \\ \theta \rho \Delta \rho +\frac{1}{2}\theta (\Delta \rho {)}^2= \\ \frac{1}{2}\theta (2\rho +\Delta \rho )\Delta \rho =\frac{1}{2}(\rho +\Delta \rho +\rho )\Delta \rho \theta = \\ \frac{1}{2}({\rho }_1+\rho )\Delta \rho \theta \end{gathered}$$

因此二重积分：

$${\int }_{y_1}^{y_2}{\int }_{x_1}^{x_2}f(x,y)dxdy={\int }_{{\theta }_1}^{{\theta }_2}{\int }_{{\rho }_0}^{{\rho }_1}f(\rho \cos \theta ,\rho \sin \theta )\rho d\rho d\theta$$