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

Table 6.5.1(a)   Calculation of center of mass via task template

Obtain $A$, the total area in region $R$ 

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

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

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

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

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

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

$\mathrm{My}\/A$ = $0$$$

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

$\mathrm{Mx}\/A$ = $\frac{4}{3\mathrm{\π}}$$$

Table 6.5.1(b)   Calculation of the centroid from first principles

Total area

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

${\∫}_{-1}^1{\∫}_0^{\sqrt{-x^2\+1}}1\ⅆy\ⅆx$

$A≔\mathrm{value}\left(q\right)$

$\frac{1}{2}\mathrm{\π}$

First Moments

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

${\∫}_{-1}^1{\∫}_0^{\sqrt{-x^2\+1}}y\ⅆy\ⅆx$

$M_x≔\mathrm{value}\left(q\right)$

$\frac{2}{3}$

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

${\∫}_{-1}^1{\∫}_0^{\sqrt{-x^2\+1}}x\ⅆy\ⅆx$

$M_y≔\mathrm{value}\left(q\right)$

$0$

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

$\left[\frac{M_y}{A}\,\frac{M_x}{A}\right]$ = $\left[0\,\frac{4}{3\mathrm{\π}}\right]$$$