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Calculus - Multivariate≻Integration≻Center of Mass≻Cartesian 2-D

Table 6.5.2(a)   Calculation of center of mass via task template

Define the density function $\mathrm{\ρ}$ 

$\mathrm{\ρ}\=x^2\+{\left(y-1\/2\right)}^2$$\stackrel{\text{assign}}{\to }$

Obtain $m$, the total mass in region $R$ 

${\int }_{-1}^1{\int }_0^{{\sqrt{1-x^2}}_{}}\mathrm{\ρ} \mathit{\ⅆ}y \mathit{\ⅆ}x$ = $\frac{3}{8}\mathrm{\π}-\frac{2}{3}$$\stackrel{\text{assign to a name}}{\to }$$m$

Obtain $M_x$, the total moments about the $x$-axis

${\int }_{-1}^1{\int }_0^{{\sqrt{1-x^2}}_{}}\mathrm{\ρ}\cdot y \mathit{\ⅆ}y \mathit{\ⅆ}x$ = $\frac{17}{30}-\frac{1}{8}\mathrm{\π}$$\stackrel{\text{assign to a name}}{\to }$$\mathrm{Mx}$

Obtain $M_y$, the total moments about the $y$-axis

${\int }_{-1}^1{\int }_0^{{\sqrt{1-x^2}}_{}}\mathrm{\ρ}\cdot x \mathit{\ⅆ}y \mathit{\ⅆ}x$ = $0$$\stackrel{\text{assign to a name}}{\to }$$\mathrm{My}$

Obtain $\stackrel{\&conjugate0;}{x}\=M_y\/m$ 

$\mathrm{My}\/m$ = $0$

Obtain $\stackrel{\&conjugate0;}{y}\=M_x\/m$ 

$\mathrm{Mx}\/m$ = $\frac{\frac{17}{30}-\frac{1}{8}\mathrm{\π}}{\frac{3}{8}\mathrm{\π}-\frac{2}{3}}$$\stackrel{\text{simplify}}{\=}$$-\frac{1}{5}\frac{-68\+15\mathrm{\π}}{9\mathrm{\π}-16}$$\stackrel{\text{at 10 digits}}{\to }$$0.3401587472$

Table 6.5.1(b)   Calculation of the center of mass from first principles

Density $\mathrm{\ρ}\left(x\,y\right)$ 

$\mathrm{\ρ}≔x^2\+{\left(y-1\/2\right)}^2\:$

Total mass

$q≔\mathrm{Int}\left(\mathrm{\ρ}\,\left[y\=0..\sqrt{1-x^2}\,x\=-1..1\right]\right)$

${\∫}_{-1}^1{\∫}_0^{\sqrt{-x^2\+1}}\left(x^2\+y^2-y\+\frac{1}{4}\right)\ⅆy\ⅆx$

$m≔\mathrm{value}\left(q\right)$

$\frac{3}{8}\mathrm{\π}-\frac{2}{3}$

First Moments

$q≔\mathrm{Int}\left(\mathrm{\ρ}\cdot y\,\left[y\=0..\sqrt{1-x^2}\,x\=-1..1\right]\right)$

${\∫}_{-1}^1{\∫}_0^{\sqrt{-x^2\+1}}\left(x^2\+{\left(y-\frac{1}{2}\right)}^2\right)y\ⅆy\ⅆx$

$M_x≔\mathrm{value}\left(q\right)$

$\frac{17}{30}-\frac{1}{8}\mathrm{\π}$

$q≔\mathrm{Int}\left(\mathrm{\ρ}\cdot x\,\left[y\=0..\sqrt{1-x^2}\,x\=-1..1\right]\right)$

${\∫}_{-1}^1{\∫}_0^{\sqrt{-x^2\+1}}\left(x^2\+{\left(y-\frac{1}{2}\right)}^2\right)x\ⅆy\ⅆx$

$M_y≔\mathrm{value}\left(q\right)$

$0$

Coordinates of center of mass $\left(\stackrel{\&conjugate0;}{x}\,\stackrel{\&conjugate0;}{y}\right)$ 

$\mathrm{CM}≔\left[\frac{M_y}{m}\,\frac{M_x}{m}\right]$

$\left[0\,\frac{\frac{17}{30}-\frac{1}{8}\mathrm{\π}}{\frac{3}{8}\mathrm{\π}-\frac{2}{3}}\right]$

$\mathrm{simplify}\left(\mathrm{CM}\right)$

$\left[0\,-\frac{1}{5}\frac{-68\+15\mathrm{\π}}{9\mathrm{\π}-16}\right]$

$\mathrm{evalf}\left(\mathrm{CM}\right)$

$\left[0.\,0.3401587474\right]$