91Ó°ÊÓ

The integrals or integral expressions in Exercises 26 represents the area of a region in the plane

${∫}_0^{2\mathrm{π}}{∫}_0^{2+\sin 4\mathrm{θ}}rdrd\mathrm{θ}$

The objective of this problem is to use polar coordinates to sketch the region and evaluate the expression

${∫}_0^{2\mathrm{π}}{∫}_0^{2+\sin 4\mathrm{θ}}rdrd\mathrm{θ}$

Here, $r=0,r=2+\sin 4\mathrm{θ}$and $\mathrm{θ}=0,\mathrm{θ}=2\mathrm{π}$

$\begin{array}{|ll|}\hline \mathrm{θ}& r=2+\sin 4\mathrm{θ}\\ 0& 1\\ \mathrm{π}/6& 2.8660\\ \mathrm{π}/4& 2.0\\ \mathrm{π}/3& 1.1339\\ \mathrm{π}/2& 2.0\\ 2\mathrm{π}/3& 2.8660\\ 3\mathrm{π}/4& 2.0\\ 5\mathrm{π}/6& 1.1339\\ \mathrm{π}& 2\\ 7\mathrm{π}/6& 2.8660\\ 5\mathrm{π}/4& 2.0\\ 3\mathrm{π}/2& 2.0\\ 7\mathrm{π}/4& 2.0\\ 2\mathrm{π}& 2.0\\ \hline\end{array}$

To sketch the region use the above table.

Plot of $r=2+\sin 4\mathrm{θ}$

The integral can be evaluated as follows:

$$\begin{gathered} {∫}_0^{2\mathrm{π}}{∫}_0^{2+\sin 4\mathrm{θ}}rdrd\mathrm{θ}={∫}_0^{2r}{\left[\frac{r^2}{2}\right]}_0^{2+\sin 4\mathrm{θ}}d\mathrm{θ} \\ {∫}_0^{2\mathrm{π}}{∫}_0^{2+\sin 4\mathrm{θ}}rdrd\mathrm{θ}={∫}_0^{2\mathrm{π}}\left[\frac{(2+\sin 4\mathrm{θ}{)}^2}{2}\right]d\mathrm{θ} \\ {∫}_0^{2\mathrm{π}}{∫}_0^{2+\sin 4\mathrm{θ}}rdrd\mathrm{θ}={∫}_0^{2n}\left[\frac{\left(4+4\sin 4\mathrm{θ}+{\sin }^{2}4\mathrm{θ}\right)}{2}\right]d\left[(a+b{)}^2=a^2+2ab+b^2\right] \\ {∫}_0^{2\mathrm{π}}{∫}_0^{2+\sin 4\mathrm{θ}}rdrd\mathrm{θ}={∫}_0^{2\mathrm{π}}\left[\frac{\{4+4\sin 4\mathrm{θ}+(1-\cos 8\mathrm{θ})/2\}}{2}d\mathrm{θ}\right. \\ {∫}_0^{2\mathrm{π}}{∫}_0^{2\sin 4\mathrm{θ}}rdrd\mathrm{θ}={∫}_0^{2\mathrm{π}}\left[\frac{\{9+8\sin 4\mathrm{θ}-\cos 8\mathrm{θ}\}}{4}\right]d\mathrm{θ} \\ \text{Integrate with respect to}\mathrm{θ}\text{.} \\ {∫}_0^{2\mathrm{π}}{∫}_0^{2+\sin 4\mathrm{θ}}rdrd\mathrm{θ}={\left[\frac{\{9\mathrm{θ}-2\cos 4\mathrm{θ}-(\sin 8\mathrm{θ})/8\}}{4}\right]}_0^{2\mathrm{π}} \\ \left[∫\sin xdx=-\cos x,∫\cos xdx=\sin x\right] \end{gathered}$$

Put the limits

$$\begin{gathered} {∫}_0^{2\mathrm{π}}{∫}_0^{2+\sin 4\mathrm{θ}}rdrd\mathrm{θ}=\left[\frac{\{18\mathrm{π}-2\cos 8\mathrm{π}-(\sin 16\mathrm{π})/8\}+2\cos \mathrm{θ}}{4}\right] \\ {∫}_0^{2e}{∫}_0^{2\sin 4\mathrm{θ}}rdrd\mathrm{θ}=\left[\frac{\{18\mathrm{π}-2\}+2}{4}\right] \\ {∫}_0^{2\mathrm{π}}{∫}_0^{2+\sin 4\mathrm{θ}}rdrd\mathrm{θ}=\frac{9\mathrm{π}}{2} \end{gathered}$$

Thus, the value of integral is

${∫}_0^{2\mathrm{π}}{∫}_0^{2\sin 4\mathrm{θ}}rdrd\mathrm{θ}=\frac{9\mathrm{π}}{2}$