First Moments

Centroid

$M_{\mathrm{yz}}\={\int }_0^2{\int }_0^4{\int }_0^{\sqrt{4-x^2}}x \mathrm{dz} \mathrm{dy} \mathrm{dx}\=\frac{32}{3}$

$\stackrel{\&conjugate0;}{x}\=\frac{M_{\mathrm{yz}}}{V}\=\frac{\frac{32}{3}}{4 \mathrm{\π}}\=\frac{8}{3 \mathrm{\π}}$

$M_{\mathrm{xz}}\={\int }_0^2{\int }_0^4{\int }_0^{\sqrt{4-x^2}}y \mathrm{dz} \mathrm{dy} \mathrm{dx}\=8 \mathrm{\π}$

$\stackrel{\&conjugate0;}{y}\=\frac{M_{\mathrm{xz}}}{V}\=\frac{8 \mathrm{\π}}{4 \mathrm{\π}}\=2$

$M_{\mathrm{xy}}\={\int }_0^2{\int }_0^4{\int }_0^{\sqrt{4-x^2}}z \mathrm{dz} \mathrm{dy} \mathrm{dx}\=\frac{32}{3}$ 

$\stackrel{\&conjugate0;}{z}\=\frac{M_{\mathrm{xy}}}{V}\=\frac{\frac{32}{3}}{4 \mathrm{\π}}\=\frac{8}{3 \mathrm{\π}}$

Table 8.3.10(a)   First moments and the coordinates of the centroid

First Moments

Center of Mass

$M_{\mathrm{yz}}\={\int }_0^2{\int }_0^4{\int }_0^{\sqrt{4-x^2}}x \left(2 x^2\+3 y^2\+4 z^2\right) \mathrm{dz} \mathrm{dy} \mathrm{dx}$ = $\frac{3584}{15}$

$\stackrel{\&conjugate0;}{x}\=\frac{M_{\mathrm{yz}}}{m}\=\frac{\frac{3584}{15}}{88 \mathrm{\π}}$ = $\frac{448}{165\mathrm{\π}}$

$M_{\mathrm{xz}}\={\int }_0^2{\int }_0^4{\int }_0^{\sqrt{4-x^2}}y \left(2 x^2\+3 y^2\+4 z^2\right) \mathrm{dz} \mathrm{dy} \mathrm{dx}$ = $240\mathrm{\π}$

$\stackrel{\&conjugate0;}{y}\=\frac{M_{\mathrm{xz}}}{m}\=\frac{240 \mathrm{\π}}{88 \mathrm{\π}}$ = $\frac{30}{11}$

$M_{\mathrm{xy}}\={\int }_0^2{\int }_0^4{\int }_0^{\sqrt{4-x^2}}z \left(2 x^2\+3 y^2\+4 z^2\right) \mathrm{dz} \mathrm{dy} \mathrm{dx}$ = $256$

$\stackrel{\&conjugate0;}{z}\=\frac{M_{\mathrm{xy}}}{m}\=\frac{256}{88 \mathrm{\π}}$ = $\frac{32}{11\mathrm{\π}}$

Table 8.3.10(b)   First moments and the coordinates of the center of mass

Tools≻Tasks≻Browse:
Calculus - Multivariate≻Integration≻Center of Mass≻Cartesian 3-D

Table 8.3.10(c)   Centroid computed by task template that implements the CenterOfMass command

Tools≻Tasks≻Browse:
Calculus - Multivariate≻Integration≻Center of Mass≻Cartesian 3-D

Table 8.3.10(d)   Centroid computed by task template that implements the CenterOfMass command

Initialize

$\mathrm{with}\left(\mathrm{Student}:-\mathrm{MultivariateCalculus}\right)\:$

Apply the CenterOfMass command from the Student MultivariateCalculus package

$\mathrm{CenterOfMass}\left(1\,z\=0..\sqrt{4-x^2}\,y\=0..4\,x\=0..2\,\mathrm{output}\=\mathrm{integral}\right)$

$\frac{{\∫}_0^2{\∫}_0^4{\∫}_0^{\sqrt{-x^2\+4}}x\ⅆz\ⅆy\ⅆx}{{\∫}_0^2{\∫}_0^4{\∫}_0^{\sqrt{-x^2\+4}}1\ⅆz\ⅆy\ⅆx}\,\frac{{\∫}_0^2{\∫}_0^4{\∫}_0^{\sqrt{-x^2\+4}}y\ⅆz\ⅆy\ⅆx}{{\∫}_0^2{\∫}_0^4{\∫}_0^{\sqrt{-x^2\+4}}1\ⅆz\ⅆy\ⅆx}\,\frac{{\∫}_0^2{\∫}_0^4{\∫}_0^{\sqrt{-x^2\+4}}z\ⅆz\ⅆy\ⅆx}{{\∫}_0^2{\∫}_0^4{\∫}_0^{\sqrt{-x^2\+4}}1\ⅆz\ⅆy\ⅆx}$

$C≔\left[\mathrm{CenterOfMass}\left(1\,z\=0..\sqrt{4-x^2}\,y\=0..4\,x\=0..2\right)\right]$

$\left[\frac{8}{3\mathrm{\π}}\,2\,\frac{8}{3\mathrm{\π}}\right]$

Table 8.3.10(e)   Centroid in Cartesian coordinates

Initialize

$\mathrm{\δ}≔2 x^2\+3 y^2\+4 z^2\:$

Apply the CenterOfMass command from the Student MultivariateCalculus package

$\mathrm{CenterOfMass}\left(\mathrm{\δ}\,z\=0..\sqrt{4-x^2}\,y\=0..4\,x\=0..2\,\mathrm{output}\=\mathrm{integral}\right)$

$\frac{{\∫}_0^2{\∫}_0^4{\∫}_0^{\sqrt{-x^2\+4}}x\left(2x^2\+3y^2\+4z^2\right)\ⅆz\ⅆy\ⅆx}{{\∫}_0^2{\∫}_0^4{\∫}_0^{\sqrt{-x^2\+4}}\left(2x^2\+3y^2\+4z^2\right)\ⅆz\ⅆy\ⅆx}\,\frac{{\∫}_0^2{\∫}_0^4{\∫}_0^{\sqrt{-x^2\+4}}y\left(2x^2\+3y^2\+4z^2\right)\ⅆz\ⅆy\ⅆx}{{\∫}_0^2{\∫}_0^4{\∫}_0^{\sqrt{-x^2\+4}}\left(2x^2\+3y^2\+4z^2\right)\ⅆz\ⅆy\ⅆx}\,\frac{{\∫}_0^2{\∫}_0^4{\∫}_0^{\sqrt{-x^2\+4}}z\left(2x^2\+3y^2\+4z^2\right)\ⅆz\ⅆy\ⅆx}{{\∫}_0^2{\∫}_0^4{\∫}_0^{\sqrt{-x^2\+4}}\left(2x^2\+3y^2\+4z^2\right)\ⅆz\ⅆy\ⅆx}$

$\mathrm{CM}≔\left[\mathrm{CenterOfMass}\left(\mathrm{\δ}\,z\=0..\sqrt{4-x^2}\,y\=0..4\,x\=0..2\right)\right]$

$\left[\frac{448}{165\mathrm{\π}}\,\frac{30}{11}\,\frac{32}{11\mathrm{\π}}\right]$

$\mathrm{evalf}\left(\mathrm{CM}\right)$ = $\left[0.8642595695\,2.727272727\,0.9259923959\right]$$$

Table 8.3.10(f)   Center of mass in Cartesian coordinates