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Center of Mass for a Planar Region in Cartesian Coordinates

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

Center of Mass for Planar Region in Cartesian Coordinates

Density:

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

Region: $\left\{u\left(x\right)\le y\le v\left(x\right),a\le x\le b\right\}$

$u\left(x\right)$

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

$v\left(x\right)$

 > ${x}$
 ${x}$ (3)

$a$

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

$b$

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

Moments$÷$Mass:

Inert Integral -

 >
 $\frac{{{∫}}_{{0}}^{{1}}{{∫}}_{{{x}}^{{2}}}^{{x}}{x}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{y}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{x}}{{{∫}}_{{0}}^{{1}}{{∫}}_{{{x}}^{{2}}}^{{x}}{1}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{y}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{x}}{,}\frac{{{∫}}_{{0}}^{{1}}{{∫}}_{{{x}}^{{2}}}^{{x}}{y}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{y}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{x}}{{{∫}}_{{0}}^{{1}}{{∫}}_{{{x}}^{{2}}}^{{x}}{1}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{y}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{x}}$ (6)

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

 > $\mathrm{Student}\left[\mathrm{MultivariateCalculus}\right]\left[\mathrm{CenterOfMass}\right]\left(,{x}=..,{y}=..\right)$
 $\frac{{1}}{{2}}{,}\frac{{2}}{{5}}$ (7)

Plot:

 > $\mathrm{Student}\left[\mathrm{MultivariateCalculus}\right]\left[\mathrm{CenterOfMass}\right]\left(,{x}=..,{y}=..,\mathrm{output}=\mathrm{plot},\mathrm{caption}=""\right)$

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