Examples

Double integral in polar

Double integral in polar

Problem: Compute ${\iint }_{\mathcal{R}}(3x+4y^2)dA$ for $\mathcal{R}$ the region with $y\ge 0$ that lies between the circles of radii 1 and 2.

Solution: We convert the integral into polar coordinates using $dA=rdrd\theta$. Obtain:

$$\begin{array}{rl}{\iint }_{\mathcal{R}}(3x+4y^2)dA& ={\int }_0^{\pi }{\int }_1^2(3r\cos (\theta )+4(r\sin (\theta ){)}^2)rdrd\theta \\ & ={\int }_0^{\pi }(7\cos (\theta )+15{\sin }^2(\theta ))d\theta =\frac{15}{2}\pi .\end{array}$$