To evaluate the double integral of a continuous function by iterated integrals over general polar regions, we consider regions analogous to Type I and Type II as discussed earlier for rectangular coordinates. It is more common to write polar equations as [latex]r = f(\theta)[/latex] than [latex]\theta = f(r)[/latex], so we describe a general polar region as [latex]R = {(r,\theta)\mid \alpha \leq \theta \leq \beta, h_1(\theta) \leq r \leq h_2(\theta) }[/latex] (see the following figure).
Figure 5. A general polar region between [latex]\alpha < 0<\beta[/latex] and [latex]h_{1}(\theta) < r<h_{2}(\theta)[/latex].
double integrals over a general polar regions
If [latex]{f}{(r,{\theta})}[/latex] is continuous on a general polar region [latex]D[/latex] as described above, then:
[latex]\underset{R}{\displaystyle\iint}f(r,\theta)r \ dr \ d\theta=\displaystyle\int_{\theta=\alpha}^{\theta=\beta}\displaystyle\int_{r=h_1(\theta)}^{r=h_2(\theta)}f(r,\theta)r \ dr \ d\theta[/latex]
Evaluate the integral [latex]\underset{D}{\displaystyle\iint}{r^2}{\sin\theta}{r} \ {dr} \ {d{\theta}}[/latex] where [latex]D[/latex] is the region bounded by the polar axis and the upper half of the cardioid [latex]{r} = {1} + {\cos\theta}[/latex].
We can describe the region [latex]D[/latex] as [latex]\{(r,\theta)\mid0\leq\theta\leq\pi,0\leq{r}\leq1+\cos\theta\}[/latex] as shown in the following figure.
Figure 6. The region [latex]D[/latex] is the top half of a cardioid.
