Question

Using polar coordinates to evaluate iterated integrals: Sketch the region determined by the limits of the given iterated integrals, and then evaluate the integrals.

${\int }_0^{\mathrm{\pi }/2}{\int }_0^3{r}^{2}drd\theta$

Short answer

${\int }_0^{\mathrm{\pi }/2}{\int }_0^3{r}^{2}drd\theta =\frac{9}{2}\mathrm{\pi }$

Step-by-step solution

Draw the region

The region determined by the limits is shown below,

Evaluate the integral

$$\begin{gathered} I={\int }_0^{\mathrm{\pi }/2}{\int }_0^3{r}^{2}drd\theta \\ I={\int }_0^{\mathrm{\pi }/2}\left[{\int }_0^3{r}^{2}dr\right]d\theta \\ I={\int }_0^{\mathrm{\pi }/2}{\left[\frac{r^3}{3}\right]}_0^{3}d\theta \\ I=9{\int }_0^{\mathrm{\pi }/2}d\theta \\ I=\frac{9}{2}\mathrm{\pi } \end{gathered}$$