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Iterated Double Integral in Polar Coordinates

 Description Compute the iterated double integral in polar coordinates.

Iterated Double Integral in Polar Coordinates

Integrand:

 > $\frac{{\mathrm{θ}}}{{1}{+}{{r}}^{{2}}}$
 $\frac{{\mathrm{θ}}}{{1}{+}{{r}}^{{2}}}$ (1)

Region: $\left\{{r}_{1}\left(\mathrm{θ}\right)\le r\le {r}_{2}\left(\mathrm{θ}\right),a\le \mathrm{θ}\le b\right\}$

${r}_{1}\left(\mathrm{θ}\right)$

 > ${0}$
 ${0}$ (2)

${r}_{2}\left(\mathrm{θ}\right)$

 > ${\mathrm{\theta }}$
 ${\mathrm{θ}}$ (3)

$a$

 > ${0}$
 ${0}$ (4)

$b$

 > $\frac{{\mathrm{\pi }}}{{3}}$
 $\frac{{1}}{{3}}{}{\mathrm{π}}$ (5)

Inert Integral:

(Note automatic insertion of Jacobian.)

 > $\mathrm{Student}\left[\mathrm{MultivariateCalculus}\right]\left[\mathrm{MultiInt}\right]\left(,{r}=..,{\mathrm{θ}}=..,\mathrm{coordinates}=\mathrm{polar}\left[{r}{,}{\mathrm{θ}}\right],\mathrm{output}=\mathrm{integral}\right)$
 ${{∫}}_{{0}}^{\frac{{1}}{{3}}{}{\mathrm{π}}}{{∫}}_{{0}}^{{\mathrm{θ}}}\frac{{\mathrm{θ}}{}{r}}{{1}{+}{{r}}^{{2}}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{r}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{\mathrm{θ}}$ (6)

Value:

 > $\mathrm{Student}\left[\mathrm{MultivariateCalculus}\right]\left[\mathrm{MultiInt}\right]\left(,{r}=..,{\mathrm{θ}}=..,\mathrm{coordinates}=\mathrm{polar}\left[{r}{,}{\mathrm{θ}}\right]\right)$
 ${-}\frac{{1}}{{2}}{}{\mathrm{ln}}{}\left({3}\right){+}\frac{{1}}{{4}}{}{\mathrm{ln}}{}\left({9}{+}{{\mathrm{π}}}^{{2}}\right){-}\frac{{1}}{{18}}{}{{\mathrm{π}}}^{{2}}{}{\mathrm{ln}}{}\left({3}\right){+}\frac{{1}}{{36}}{}{{\mathrm{π}}}^{{2}}{}{\mathrm{ln}}{}\left({9}{+}{{\mathrm{π}}}^{{2}}\right){-}\frac{{1}}{{36}}{}{{\mathrm{π}}}^{{2}}$ (7)

Stepwise Evaluation:

 > $\mathrm{Student}\left[\mathrm{MultivariateCalculus}\right]\left[\mathrm{MultiInt}\right]\left(,{r}=..,{\mathrm{θ}}=..,\mathrm{coordinates}=\mathrm{polar}\left[{r}{,}{\mathrm{θ}}\right],\mathrm{output}=\mathrm{steps}\right)$
 $\begin{array}{ccc}\multicolumn{3}{c}{{{\int }}_{{0}}^{\frac{{\mathrm{\pi }}}{{3}}}{{\int }}_{{0}}^{{\mathrm{\theta }}}\frac{{\mathrm{\theta }}{}{r}}{{1}{+}{{r}}^{{2}}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{r}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{\mathrm{\theta }}}\\ \phantom{\rule[-0.0ex]{5.0em}{0.0ex}}& {\text{=}}& {{\int }}_{{0}}^{\frac{{\mathrm{\pi }}}{{3}}}\left(\genfrac{}{}{0}{}{\frac{{\mathrm{\theta }}{}{\mathrm{ln}}{}\left({1}{+}{{r}}^{{2}}\right)}{{2}}}{\phantom{{r}{=}{0}{..}{\mathrm{\theta }}}}{|}\genfrac{}{}{0}{}{\phantom{\frac{{\mathrm{\theta }}{}{\mathrm{ln}}{}\left({1}{+}{{r}}^{{2}}\right)}{{2}}}}{{r}{=}{0}{..}{\mathrm{\theta }}}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{\mathrm{\theta }}\hfill \\ \phantom{\rule[-0.0ex]{5.0em}{0.0ex}}& {\text{=}}& {{\int }}_{{0}}^{\frac{{\mathrm{\pi }}}{{3}}}\frac{{\mathrm{ln}}{}\left({1}{+}{{\mathrm{\theta }}}^{{2}}\right){}{\mathrm{\theta }}}{{2}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{\mathrm{\theta }}\hfill \\ \phantom{\rule[-0.0ex]{5.0em}{0.0ex}}& {\text{=}}& \genfrac{}{}{0}{}{\left(\frac{\left({1}{+}{{\mathrm{\theta }}}^{{2}}\right){}{\mathrm{ln}}{}\left({1}{+}{{\mathrm{\theta }}}^{{2}}\right)}{{4}}{-}\frac{{1}}{{4}}{-}\frac{{{\mathrm{\theta }}}^{{2}}}{{4}}\right)}{\phantom{{\mathrm{\theta }}{=}{0}{..}\frac{{\mathrm{\pi }}}{{3}}}}{|}\genfrac{}{}{0}{}{\phantom{\left(\frac{\left({1}{+}{{\mathrm{\theta }}}^{{2}}\right){}{\mathrm{ln}}{}\left({1}{+}{{\mathrm{\theta }}}^{{2}}\right)}{{4}}{-}\frac{{1}}{{4}}{-}\frac{{{\mathrm{\theta }}}^{{2}}}{{4}}\right)}}{{\mathrm{\theta }}{=}{0}{..}\frac{{\mathrm{\pi }}}{{3}}}\hfill \end{array}$
 ${-}\frac{{1}}{{2}}{}{\mathrm{ln}}{}\left({3}\right){+}\frac{{1}}{{4}}{}{\mathrm{ln}}{}\left({9}{+}{{\mathrm{π}}}^{{2}}\right){-}\frac{{1}}{{18}}{}{{\mathrm{π}}}^{{2}}{}{\mathrm{ln}}{}\left({3}\right){+}\frac{{1}}{{36}}{}{{\mathrm{π}}}^{{2}}{}{\mathrm{ln}}{}\left({9}{+}{{\mathrm{π}}}^{{2}}\right){-}\frac{{1}}{{36}}{}{{\mathrm{π}}}^{{2}}$ (8)

 Commands Used