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

Center of Mass for 3D Region in Cartesian Coordinates

 Description Determine $\stackrel{&conjugate0;}{x}$, $\stackrel{&conjugate0;}{y}$, and $\stackrel{&conjugate0;}{z}$, the center of mass  coordinates for a 3D region in Cartesian coordinates .

Center of Mass for 3D Region in Cartesian Coordinates

Density:

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



Region: $\left\{{z}_{1}\left(x,y\right)\le z\le {z}_{2}\left(x,y\right),{y}_{1}\left(x\right)\le y\le {y}_{2}\left(x\right),a\le x\le b\right\}$

${z}_{1}\left(x,y\right)$

 >
 ${1}{-}{x}{-}{y}$ (2)

${z}_{2}\left(x,y\right)$

 > ${10}{-}{{x}}^{{2}}{-}{{y}}^{{2}}$
 ${10}{-}{{x}}^{{2}}{-}{{y}}^{{2}}$ (3)

${y}_{1}\left(x\right)$

 > ${{x}}^{{2}}$
 ${{x}}^{{2}}$ (4)

${y}_{2}\left(x\right)$

 > ${x}$
 ${x}$ (5)

$a$

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

$b$

 > ${1}$
 ${1}$ (7)

Moments$÷$Mass:

Inert Integral -

 >
 $\frac{{{∫}}_{{0}}^{{1}}{{∫}}_{{{x}}^{{2}}}^{{x}}{{∫}}_{{1}{-}{x}{-}{y}}^{{10}{-}{{x}}^{{2}}{-}{{y}}^{{2}}}{{z}}^{{2}}{}{x}{}{y}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{z}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{y}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{x}}{{{∫}}_{{0}}^{{1}}{{∫}}_{{{x}}^{{2}}}^{{x}}{{∫}}_{{1}{-}{x}{-}{y}}^{{10}{-}{{x}}^{{2}}{-}{{y}}^{{2}}}{x}{}{y}{}{z}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{z}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{y}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{x}}{,}\frac{{{∫}}_{{0}}^{{1}}{{∫}}_{{{x}}^{{2}}}^{{x}}{{∫}}_{{1}{-}{x}{-}{y}}^{{10}{-}{{x}}^{{2}}{-}{{y}}^{{2}}}{{y}}^{{2}}{}{x}{}{z}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{z}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{y}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{x}}{{{∫}}_{{0}}^{{1}}{{∫}}_{{{x}}^{{2}}}^{{x}}{{∫}}_{{1}{-}{x}{-}{y}}^{{10}{-}{{x}}^{{2}}{-}{{y}}^{{2}}}{x}{}{y}{}{z}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{z}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{y}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{x}}{,}\frac{{{∫}}_{{0}}^{{1}}{{∫}}_{{{x}}^{{2}}}^{{x}}{{∫}}_{{1}{-}{x}{-}{y}}^{{10}{-}{{x}}^{{2}}{-}{{y}}^{{2}}}{{x}}^{{2}}{}{y}{}{z}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{z}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{y}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{x}}{{{∫}}_{{0}}^{{1}}{{∫}}_{{{x}}^{{2}}}^{{x}}{{∫}}_{{1}{-}{x}{-}{y}}^{{10}{-}{{x}}^{{2}}{-}{{y}}^{{2}}}{x}{}{y}{}{z}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{z}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{y}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{x}}$ (8)

Explicit values for $\stackrel{&conjugate0;}{x}$, $\stackrel{&conjugate0;}{y}$, and $\stackrel{&conjugate0;}{z}$

 >
 $\frac{{3507743}}{{573782}}{,}\frac{{332885}}{{573782}}{,}\frac{{2494317}}{{3729583}}$ (9)

 > 
 Commands Used