Cylindrical - Maple Help

Iterated Triple Integral in Cylindrical Coordinates

 Description Compute the iterated triple integral in cylindrical coordinates.

Iterated Triple Integral in Cylindrical Coordinates

Integrand:

 > ${z}$
 ${z}$ (1)

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

${z}_{1}\left(r,\mathrm{θ}\right)$

 > ${r}$
 ${r}$ (2)

${z}_{2}\left(r,\mathrm{θ}\right)$

 > ${1}$
 ${1}$ (3)

${r}_{1}\left(\mathrm{\theta }\right)$

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

${r}_{2}\left(\mathrm{θ}\right)$

 > ${1}$
 ${1}$ (5)

$a$

 > ${0}$
 ${0}$ (6)

$b$

 >
 ${2}{}{\mathrm{π}}$ (7)

Inert Integral:

(Note automatic insertion of Jacobian.)

 > $\mathrm{Student}\left[\mathrm{MultivariateCalculus}\right]\left[\mathrm{MultiInt}\right]\left(,{z}=..,{r}=..,{\mathrm{θ}}=..,\mathrm{coordinates}=\mathrm{cylindrical}\left[{r}{,}{\mathrm{θ}}{,}{z}\right],\mathrm{output}=\mathrm{integral}\right)$
 ${{∫}}_{{0}}^{{2}{}{\mathrm{π}}}{{∫}}_{{0}}^{{1}}{{∫}}_{{r}}^{{1}}{z}{}{r}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{z}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{r}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{\mathrm{θ}}$ (8)

Value:

 > $\mathrm{Student}\left[\mathrm{MultivariateCalculus}\right]\left[\mathrm{MultiInt}\right]\left(,{z}=..,{r}=..,{\mathrm{θ}}=..,\mathrm{coordinates}=\mathrm{cylindrical}\left[{r}{,}{\mathrm{θ}}{,}{z}\right]\right)$
 $\frac{{1}}{{4}}{}{\mathrm{π}}$ (9)

Stepwise Evaluation:

 > $\mathrm{Student}\left[\mathrm{MultivariateCalculus}\right]\left[\mathrm{MultiInt}\right]\left(,{z}=..,{r}=..,{\mathrm{θ}}=..,\mathrm{coordinates}=\mathrm{cylindrical}\left[{r}{,}{\mathrm{θ}}{,}{z}\right],\mathrm{output}=\mathrm{steps}\right)$
 $\begin{array}{ccc}\multicolumn{3}{c}{{{\int }}_{{0}}^{{2}{}{\mathrm{\pi }}}{{\int }}_{{0}}^{{1}}{{\int }}_{{r}}^{{1}}{z}{}{r}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{z}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{r}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{\mathrm{\theta }}}\\ \phantom{\rule[-0.0ex]{5.0em}{0.0ex}}& {\text{=}}& {{\int }}_{{0}}^{{2}{}{\mathrm{\pi }}}{{\int }}_{{0}}^{{1}}\left(\genfrac{}{}{0}{}{\frac{{{z}}^{{2}}{}{r}}{{2}}}{\phantom{{z}{=}{r}{..}{1}}}{|}\genfrac{}{}{0}{}{\phantom{\frac{{{z}}^{{2}}{}{r}}{{2}}}}{{z}{=}{r}{..}{1}}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{r}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{\mathrm{\theta }}\hfill \\ \phantom{\rule[-0.0ex]{5.0em}{0.0ex}}& {\text{=}}& {{\int }}_{{0}}^{{2}{}{\mathrm{\pi }}}{{\int }}_{{0}}^{{1}}\frac{{r}{}\left({1}{-}{{r}}^{{2}}\right)}{{2}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{r}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{\mathrm{\theta }}\hfill \\ \phantom{\rule[-0.0ex]{5.0em}{0.0ex}}& {\text{=}}& {{\int }}_{{0}}^{{2}{}{\mathrm{\pi }}}\left(\genfrac{}{}{0}{}{\left({-}\frac{{1}}{{8}}{}{{r}}^{{4}}{+}\frac{{1}}{{4}}{}{{r}}^{{2}}\right)}{\phantom{{r}{=}{0}{..}{1}}}{|}\genfrac{}{}{0}{}{\phantom{\left({-}\frac{{1}}{{8}}{}{{r}}^{{4}}{+}\frac{{1}}{{4}}{}{{r}}^{{2}}\right)}}{{r}{=}{0}{..}{1}}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{\mathrm{\theta }}\hfill \\ \phantom{\rule[-0.0ex]{5.0em}{0.0ex}}& {\text{=}}& {{\int }}_{{0}}^{{2}{}{\mathrm{\pi }}}\frac{{1}}{{8}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{\mathrm{\theta }}\hfill \\ \phantom{\rule[-0.0ex]{5.0em}{0.0ex}}& {\text{=}}& \genfrac{}{}{0}{}{\frac{{\mathrm{\theta }}}{{8}}}{\phantom{{\mathrm{\theta }}{=}{0}{..}{2}{}{\mathrm{\pi }}}}{|}\genfrac{}{}{0}{}{\phantom{\frac{{\mathrm{\theta }}}{{8}}}}{{\mathrm{\theta }}{=}{0}{..}{2}{}{\mathrm{\pi }}}\hfill \end{array}$
 $\frac{{1}}{{4}}{}{\mathrm{π}}$ (10)

 Commands Used