91Ó°ÊÓ

The objective is to show that,

${∫}_0^{\mathrm{π}/3}{∫}_0^{1+ses\mathrm{θ}}rdrd\mathrm{θ}=\frac{1}{4}\mathrm{π}+\frac{9}{16}\sqrt{3}$

and

${∫}_{\mathrm{π}/3}^{\mathrm{π}/2}{∫}_0^{3\cos \mathrm{θ}}rdrd\mathrm{θ}=\frac{3}{8}\mathrm{π}-\frac{9}{16}\sqrt{3}$

${∫}_0^{\mathrm{π}/3}{∫}_0^{1+\cos \mathrm{θ}}rdrd\mathrm{θ}$

Integrate with respect to $r$.

${∫}_0^{\mathrm{π}/3}{∫}_0^{1+ces\mathrm{θ}}rdrd\mathrm{θ}={∫}_0^{\mathrm{π}/3}{\left[\frac{r^2}{2}\right]}_0^{1+\cos \mathrm{θ}}d\mathrm{θ}$

Substitute the limits

${∫}_0^{\mathrm{π}/3}{∫}_0^{1+ses\mathrm{θ}}rdrd\mathrm{θ}={∫}_0^{\mathrm{π}/3}\left[\frac{(1+\cos \mathrm{θ}{)}^2-0}{2}\right]d\mathrm{θ}$

${∫}_0^{\mathrm{π}/3}{∫}_0^{1+\cos \mathrm{θ}}rdrd\mathrm{θ}={∫}_0^{\mathrm{π}/3}\left[\frac{\left(1+2\cos \mathrm{θ}+{\cos }^2\mathrm{θ}\right)}{2}\right]d\mathrm{θ}\left[(a+b{)}^2=a^2+2ab+b^2\right]$

${∫}_0^{\mathrm{π}/3}{∫}_0^{1+\cos \mathrm{θ}}rdrd\mathrm{θ}={∫}_0^{\mathrm{π}/3}\left[\frac{\{1+2\cos \mathrm{θ}+(1+\cos 2\mathrm{θ})/2\}}{2}\right]\mathrm{θ}$

${∫}_0^{\mathrm{π}/3}{∫}_0^{1+\cos \mathrm{θ}}rdrd\mathrm{θ}={∫}_0^{\mathrm{π}/3}\left[\frac{3+4\cos \mathrm{θ}+\cos 2\mathrm{θ}}{4}\right]d\mathrm{θ}$

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

${∫}_0^{\mathrm{π}/3}{∫}_0^{1+\cos \mathrm{θ}}rdrd\mathrm{θ}={\left[\frac{3\mathrm{θ}+4\sin \mathrm{θ}+\left(\sin 2\mathrm{θ}\right)/2}{4}\right]}_0^{*/3}\left[∫\cos xdx=\sin x\right]$

Put the limits

${∫}_0^{\mathrm{π}/3}{∫}_0^{1+\cos \mathrm{θ}}rdrd\mathrm{θ}=\left[\frac{\mathrm{π}+4\sin (\mathrm{π}/3)+(\sin 2\mathrm{π}/3)/2-0}{4}\right]$

${∫}_0^{z/3}{∫}_0^{1+ces\mathrm{θ}}rdrd\mathrm{θ}=\left[\frac{\mathrm{π}+2\sqrt{3}+\sqrt{3}/4}{4}\right]$

${∫}_0^{\mathrm{π}/3}{∫}_0^{1+\cos \mathrm{θ}}rdrd\mathrm{θ}=\frac{\mathrm{π}}{4}+\frac{9}{16}\sqrt{3}$

${∫}_{\mathrm{π}/3}^{\mathrm{π}/2}{∫}_0^{3\cos \mathrm{θ}}rdrd\mathrm{θ}$

Integrate with respect to$r$

${∫}_{\mathrm{π}/3}^{\mathrm{π}/2}{∫}_0^{sen\mathrm{θ}}rdrd\mathrm{θ}={∫}_{\mathrm{π}/3}^{\mathrm{π}/2}{\left[\frac{r^2}{2}\right]}_0^{3sen\mathrm{θ}}d\mathrm{θ}$

Substitute the limits

${∫}_{\mathrm{π}/3}^{\mathrm{π}/2}{∫}_0^{3\cos \mathrm{θ}}rdrd\mathrm{θ}={∫}_{\mathrm{π}/3}^{\mathrm{π}/2}\left[\frac{(3\cos \mathrm{θ}{)}^2-0}{2}\right]d\mathrm{θ}$

${∫}_{\mathrm{π}/3}^{\mathrm{π}/2}{∫}_0^{3\cos \mathrm{θ}}rdrd\mathrm{θ}={∫}_{\mathrm{π}/3}^{\mathrm{π}/2}\left[\frac{9{\cos }^2\mathrm{θ}}{2}\right]d\mathrm{θ}$

${∫}_{\mathrm{π}/3}^{\mathrm{π}/2}{∫}_0^{3\cos \mathrm{θ}}rdrd\mathrm{θ}=\frac{9}{4}{∫}_{\mathrm{π}/3}^{\mathrm{π}/2}[1+\cos 2\mathrm{θ}]d\mathrm{θ}\left[{\cos }^2\mathrm{θ}=\frac{1}{2}(1+\cos 2\mathrm{θ})\right]$

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

${∫}_{\mathrm{π}/3}^{\mathrm{π}/2}{∫}_0^{3\cos \mathrm{θ}}rdrd\mathrm{θ}=\frac{9}{4}{\left[\mathrm{θ}+\frac{\sin 2\mathrm{θ}}{2}\right]}_{x/3}^{x/2}$

Substitute the limits

${∫}_{\mathrm{π}/3}^{\mathrm{π}/2}{∫}_0^{3\cos \mathrm{θ}}rdrd\mathrm{θ}=\frac{9}{4}\left[\left(\frac{\mathrm{π}}{2}+\frac{\sin \mathrm{π}}{2}\right)-\left(\frac{\mathrm{π}}{3}+\frac{\sin (2\mathrm{π}/3)}{2}\right)\right]$

${∫}_{\mathrm{π}/3}^{\mathrm{π}/2}{∫}_0^{3\cos \mathrm{θ}}rdrd\mathrm{θ}=\frac{9}{4}\left[\left(\frac{\mathrm{π}}{2}\right)-\left(\frac{\mathrm{π}}{3}+\frac{\sqrt{3}}{4}\right)\right]$

role="math" localid="1650520240502" ${∫}_{\mathrm{π}/3}^{\mathrm{π}/2}{∫}_0^{3\cos \mathrm{θ}}rdrd\mathrm{θ}=\frac{3\mathrm{π}}{8}-\frac{9}{16}\sqrt{3}$

role="math" localid="1650520257012" ${∫}_{\mathrm{π}/3}^{\mathrm{π}/2}{∫}_0^{3c\mathrm{os}\mathrm{θ}}rdrd\mathrm{θ}=\frac{3\mathrm{π}}{8}-\frac{9}{16}\sqrt{3}$