Center of Mass for 3-D Region in Cylindrical Coordinates - Maple Help

Center of Mass for 3-D Region in Cylindrical Coordinates

 Description Determine $\stackrel{&conjugate0;}{r}$, $\stackrel{&conjugate0;}{\mathrm{\theta }}$, and $\stackrel{&conjugate0;}{z}$, the center of mass  coordinates for a 3-D region in cylindrical coordinates .

Center of Mass for 3-D Region in Cylindrical Coordinates

Density:

 > $z$
 ${z}$ (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)

${r}_{1}\left(\mathrm{θ}\right)$

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

${r}_{2}\left(\mathrm{θ}\right)$

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

$a$

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

$b$

 > $\frac{{\mathrm{\pi }}}{{3}}$
 $\frac{{1}}{{3}}{}{\mathrm{π}}$ (7)

Moments ÷ Mass:

Inert Integral -

 > $\mathrm{Student}\left[\mathrm{MultivariateCalculus}\right]\left[\mathrm{CenterOfMass}\right]\left(,z=..,r=..,\mathrm{θ}=..,\mathrm{coordinates}=\mathrm{cylindrical}\left[r,\mathrm{θ},z\right],\mathrm{output}=\mathrm{integral}\right)$
 $\frac{{{∫}}_{{0}}^{\frac{{1}}{{3}}{}{\mathrm{π}}}{{∫}}_{{0}}^{{1}}{{∫}}_{{r}}^{{1}}{\mathrm{cos}}{}\left({\mathrm{θ}}\right){}{{r}}^{{2}}{}{z}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{z}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{r}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{\mathrm{θ}}}{{{∫}}_{{0}}^{\frac{{1}}{{3}}{}{\mathrm{π}}}{{∫}}_{{0}}^{{1}}{{∫}}_{{r}}^{{1}}{r}{}{z}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{z}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{r}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{\mathrm{θ}}}{,}\frac{{{∫}}_{{0}}^{\frac{{1}}{{3}}{}{\mathrm{π}}}{{∫}}_{{0}}^{{1}}{{∫}}_{{r}}^{{1}}{\mathrm{sin}}{}\left({\mathrm{θ}}\right){}{{r}}^{{2}}{}{z}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{z}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{r}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{\mathrm{θ}}}{{{∫}}_{{0}}^{\frac{{1}}{{3}}{}{\mathrm{π}}}{{∫}}_{{0}}^{{1}}{{∫}}_{{r}}^{{1}}{r}{}{z}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{z}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{r}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{\mathrm{θ}}}{,}\frac{{{∫}}_{{0}}^{\frac{{1}}{{3}}{}{\mathrm{π}}}{{∫}}_{{0}}^{{1}}{{∫}}_{{r}}^{{1}}{{z}}^{{2}}{}{r}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{z}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{r}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{\mathrm{θ}}}{{{∫}}_{{0}}^{\frac{{1}}{{3}}{}{\mathrm{π}}}{{∫}}_{{0}}^{{1}}{{∫}}_{{r}}^{{1}}{r}{}{z}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{z}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{r}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{\mathrm{θ}}}$ (8)

Explicit values for $\stackrel{&conjugate0;}{r}$, $\stackrel{&conjugate0;}{\mathrm{\theta }}$, and $\stackrel{&conjugate0;}{z}$, the center of mass given in cylindrical coordinates:

 > $\mathrm{Student}\left[\mathrm{MultivariateCalculus}\right]\left[\mathrm{CenterOfMass}\right]\left(,z=..,r=..,\mathrm{θ}=..,\mathrm{coordinates}=\mathrm{cylindrical}\left[r,\mathrm{\theta },z\right]\right)$
 $\frac{{8}}{{5}{}{\mathrm{π}}}{,}\frac{{1}}{{6}}{}{\mathrm{π}}{,}\frac{{4}}{{5}}$ (9)

 Commands Used