First Moments

Centroid

$M_{\mathrm{yz}}\={\int }_0^1{\int }_0^{2-x}{\int }_0^{2-x-y}x \mathrm{dz} \mathrm{dy} \mathrm{dx}\=\frac{11}{24}$

$\stackrel{\&conjugate0;}{x}\=\frac{M_{\mathrm{yz}}}{V}\=\frac{\frac{11}{24}}{\frac{7}{6}}\=\frac{11}{28}$

$M_{\mathrm{xz}}\={\int }_0^1{\int }_0^{2-x}{\int }_0^{2-x-y}y \mathrm{dz} \mathrm{dy} \mathrm{dx}\=\frac{5}{8}$

$\stackrel{\&conjugate0;}{y}\=\frac{M_{\mathrm{xz}}}{V}\=\frac{\frac{5}{8}}{\frac{7}{6}}\=\frac{15}{28}$

$M_{\mathrm{xy}}\={\int }_0^1{\int }_0^{2-x}{\int }_0^{2-x-y}z \mathrm{dz} \mathrm{dy} \mathrm{dx}\=\frac{5}{8}$ 

$\stackrel{\&conjugate0;}{z}\=\frac{M_{\mathrm{xy}}}{V}\=\frac{\frac{5}{8}}{\frac{7}{6}}\=\frac{15}{28}$

Table 8.3.12(a)   First moments and the coordinates of the centroid

First Moments

Center of Mass

$M_{\mathrm{yz}}\={\int }_0^1{\int }_0^{2-x}{\int }_0^{2-x-y}{x}^2 y^2{z}^3 \mathrm{dz} \mathrm{dy} \mathrm{dx}$ = $\frac{121}{18900}$

$\stackrel{\&conjugate0;}{x}\=\frac{M_{\mathrm{yz}}}{m}\=\frac{\frac{121}{18900}}{\frac{251}{15120}}$ = $\frac{484}{1255}$

$M_{\mathrm{xz}}\={\int }_0^1{\int }_0^{2-x}{\int }_0^{2-x-y}x y^3{z}^3 \mathrm{dz} \mathrm{dy} \mathrm{dx}$ = $\frac{1013}{100800}$

$\stackrel{\&conjugate0;}{y}\=\frac{M_{\mathrm{xz}}}{m}\=\frac{\frac{1013}{100800}}{\frac{251}{15120}}$ = $\frac{3039}{5020}$

$M_{\mathrm{xy}}\={\int }_0^1{\int }_0^{2-x}{\int }_0^{2-x-y}x y^2{z}^4 \mathrm{dz} \mathrm{dy} \mathrm{dx}$ = $\frac{1013}{75600}$

$\stackrel{\&conjugate0;}{z}\=\frac{M_{\mathrm{xy}}}{m}\=\frac{\frac{1013}{75600}}{\frac{251}{15120}}$ = $\frac{1013}{1255}$

Table 8.3.12(b)   First moments and the coordinates of the center of mass

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

Table 8.3.12(c)   Centroid computed by task template that implements the CenterOfMass command

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

Table 8.3.12(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

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

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

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

$\left[\frac{11}{28}\,\frac{15}{28}\,\frac{15}{28}\right]$

Table 8.3.12(e)   Centroid in Cartesian coordinates

Initialize

$\mathrm{\δ}≔x y^2{z}^3\:$

Apply the CenterOfMass command from the Student MultivariateCalculus package

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

$\frac{{\∫}_0^1{\∫}_0^{2-x}{\∫}_0^{2-x-y}{x}^2{y}^2{z}^3\ⅆz\ⅆy\ⅆx}{{\∫}_0^1{\∫}_0^{2-x}{\∫}_0^{2-x-y}x{y}^2{z}^3\ⅆz\ⅆy\ⅆx}\,\frac{{\∫}_0^1{\∫}_0^{2-x}{\∫}_0^{2-x-y}{y}^{3}x{z}^3\ⅆz\ⅆy\ⅆx}{{\∫}_0^1{\∫}_0^{2-x}{\∫}_0^{2-x-y}x{y}^2{z}^3\ⅆz\ⅆy\ⅆx}\,\frac{{\∫}_0^1{\∫}_0^{2-x}{\∫}_0^{2-x-y}{z}^{4}x{y}^2\ⅆz\ⅆy\ⅆx}{{\∫}_0^1{\∫}_0^{2-x}{\∫}_0^{2-x-y}x{y}^2{z}^3\ⅆz\ⅆy\ⅆx}$

$\mathrm{CM}≔\left[\mathrm{CenterOfMass}\left(\mathrm{\δ}\,z\=0..2-x-y\,y\=0..2-x\,x\=0..1\right)\right]$

$\left[\frac{484}{1255}\,\frac{3039}{5020}\,\frac{1013}{1255}\right]$

$\mathrm{evalf}\left(\mathrm{CM}\right)$ = $\left[0.3856573705\,0.6053784861\,0.8071713147\right]$$$

Table 8.3.12(f)   Center of mass in Cartesian coordinates