91Ó°ÊÓ

The expression is

$2{∫}_0^{2\mathrm{π}/3}{∫}_0^{(1/2)+\cos \mathrm{θ}}rdrd\mathrm{θ}-2{∫}_{\mathrm{π}}^{4\mathrm{π}/3}{∫}_0^{(1/2)+\cos \mathrm{θ}}rdrd\mathrm{θ}$

Here, $r=0,r=(1/2)+\cos \mathrm{θ}$and $\mathrm{θ}=0,\mathrm{θ}=2\mathrm{π}/3,\mathrm{θ}=\mathrm{π}$and $\mathrm{θ}=4\mathrm{π}/3$

$\begin{array}{|ll|}\hline \mathrm{θ}& r=(1/2)+\cos \mathrm{θ}\\ 0& 1.5\\ \mathrm{π}/4& 1.2071\\ \mathrm{π}/2& 0.5\\ 3\mathrm{π}/4& -0.2071\\ \mathrm{π}& -0.5\\ 5\mathrm{π}/4& -0.2071\\ 4\mathrm{π}/3& 0\\ \hline\end{array}$

To sketch the region use the above table

Plot of $r=(1/2)+\cos \mathrm{θ}$

$I=2{∫}_0^{2\mathrm{π}/3}{∫}_0^{(1/2)+\cos \mathrm{θ}}rdrd\mathrm{θ}-2{∫}_{\mathrm{π}}^{4\mathrm{π}/3}{∫}_0^{(1/2)+sen\mathrm{θ}}rdrd\mathrm{θ}$

$I=I_1-I_2$

Where,

$$\begin{gathered} I_1=2{∫}_0^{2\mathrm{π}/3}{∫}_0^{(1/2)+eos\mathrm{θ}}rdrd\mathrm{θ} \\ {I}_1=2{∫}_0^{2\mathrm{π}/3}{\left[\frac{r^2}{2}\right]}_0^{(1/2)+\cos \mathrm{θ}}d\mathrm{θ} \\ {I}_1=2{∫}_0^{2\mathrm{π}/3}\left[\frac{\left\{\right(1/2)+\cos \mathrm{θ}{\}}^2-0}{2}d\mathrm{θ}\right. \\ {I}_1=2{∫}_0^{2\mathrm{π}/3}\left[\frac{\left\{\frac{1}{4}+\cos \mathrm{θ}+{\cos }^2\mathrm{θ}\right\}}{2}\right]\mathrm{θ} \\ {I}_1=2{∫}_0^{2\mathrm{π}/3}\left[\frac{\left\{\frac{1}{4}+\cos \mathrm{θ}+\frac{1}{2}(1+\cos 2\mathrm{θ})\right\}}{2}d\mathrm{θ}\right. \end{gathered}$$

Integrate with respect to $\mathrm{θ}$.

$$\begin{gathered} I_1=2{\left[\frac{\left\{\frac{\mathrm{θ}}{4}+\sin \mathrm{θ}+\frac{1}{2}\left(\mathrm{θ}+\frac{1}{2}\sin 2\mathrm{θ}\right)\right\}}{2}\right]}_0^{2\mathrm{π}/3} \\ {I}_1=2\left[\frac{\left.\left\{\frac{3}{4}·\frac{2\mathrm{π}}{3}+\sin \left(\frac{2\mathrm{π}}{3}\right)+\frac{1}{4}\sin \left(\frac{4\mathrm{π}}{3}\right)\right\}-\left\{0\right\}\right]}{2}\right] \\ {I}_1=2\left[\frac{\left\{\frac{\mathrm{π}}{2}+\frac{\sqrt{3}}{2}-\frac{\sqrt{3}}{8}\right\}}{2}\right] \\ {I}_1=2\left[\frac{\left\{\frac{\mathrm{π}}{2}+\frac{3\sqrt{3}}{8}\right\}}{2}\right] \\ {I}_1=\frac{\mathrm{π}}{2}+\frac{3\sqrt{3}}{8} \end{gathered}$$

Now

$$\begin{gathered} I_2=2{∫}_{\mathrm{π}}^{4\mathrm{π}/3}{∫}_0^{(1/2)+\cos \mathrm{θ}}rdrd\mathrm{θ} \\ {I}_2=2{∫}_{\mathrm{π}}^{4\mathrm{π}/3}{\left[\frac{r^2}{2}\right]}_0^{(1/2)+\cos \mathrm{θ}}d\mathrm{θ} \\ {I}_2=2{∫}_n^{4\mathrm{π}/3}\left[\left\{\frac{\left.\frac{1}{4}+\cos \mathrm{θ}+{\cos }^2\mathrm{θ}\right\}}{2}\right]d\mathrm{θ}\right. \\ {I}_2=2{∫}_n^{4\mathrm{π}/3}\left[\frac{\left\{\frac{1}{4}+\cos \mathrm{θ}+\frac{1}{2}(1+\cos 2\mathrm{θ})\right\}}{2}d\mathrm{θ}\right. \\ {I}_2=2{\left[\frac{\left\{\frac{\mathrm{θ}}{4}+\sin \mathrm{θ}+\frac{1}{2}\left(\mathrm{θ}+\frac{1}{2}\sin 2\mathrm{θ}\right)\right\}}{2}\right]}_n^{4n} \\ {I}_2=2\left[\frac{\left\{\frac{1}{4}·\frac{4\mathrm{π}}{3}+\sin \left(\frac{4\mathrm{π}}{3}\right)+\frac{1}{4}\sin \left(\frac{8\mathrm{π}}{3}\right)\right\}-\left\{\frac{\mathrm{π}}{4}+\sin \mathrm{π}+\frac{1}{2}\left(\mathrm{π}+\frac{1}{2}\sin 2\mathrm{π}\right)\right\}}{2}\right] \\ {I}_2=2\left[\frac{\left\{\frac{\mathrm{π}}{3}-\frac{\sqrt{3}}{2}+\frac{1}{4},\frac{\sqrt{3}}{2}\right\}-\left\{\frac{\mathrm{π}}{4}+\frac{\mathrm{π}}{2}\right\}}{2}\right] \\ {I}_2=2\left[\frac{\left\{\frac{\mathrm{π}}{3}-\frac{\sqrt{3}}{2}+\frac{1}{4}·\frac{\sqrt{3}}{2}\right\}-\left\{\frac{\mathrm{π}}{4}+\frac{\mathrm{π}}{2}\right\}}{2}\right] \\ {I}_2=-\frac{5\mathrm{π}}{12}-\frac{3\sqrt{3}}{8} \end{gathered}$$

Therefore,

$$\begin{gathered} I=\frac{\mathrm{Ï€}}{2}+\frac{3\sqrt{3}}{8}-\frac{5\mathrm{Ï€}}{12}-\frac{3\sqrt{3}}{8} \\ I=\frac{\mathrm{Ï€}}{12} \end{gathered}$$

Thus, the value of integral is

$2{∫}_0^{2\mathrm{π}/3}{∫}_0^{(1/2)+ces\mathrm{θ}}rdrd\mathrm{θ}-2{∫}_{\mathrm{π}}^{4\mathrm{π}/3}{∫}_0^{(1/2)res\mathrm{θ}}rdrd\mathrm{θ}=\frac{\mathrm{π}}{12}$