Obtain $A$, the total area of the region

${\int }_0^1{\int }_0^{y-y^3}1 \mathit{\ⅆ}x \mathit{\ⅆ}y$ = $\frac{1}{4}$$\stackrel{\text{assign to a name}}{\to }$$A$ 

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

${\int }_0^1{\int }_0^{y-y^3}y \mathit{\ⅆ}x \mathit{\ⅆ}y$ = $\frac{2}{15}$$\stackrel{\text{assign to a name}}{\to }$$\mathrm{Mx}$$$ 

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

${\int }_0^1{\int }_0^{y-y^3}x \mathit{\ⅆ}x \mathit{\ⅆ}y$ = $\frac{4}{105}$$\stackrel{\text{assign to a name}}{\to }$$\mathrm{My}$$$

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

$\mathrm{My}\/A$ = $\frac{16}{105}$$$

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

$\mathrm{Mx}\/A$ = $\frac{8}{15}$$$

Table 6.5.9(a)   Calculation of the centroid from first principles

Obtain the total area of the region

$q≔\mathrm{Int}\left(1\,\left[x\=0..y-y^3\,y\=0..1\right]\right)$

${\∫}_0^1{\∫}_0^{-y^3\+y}1\ⅆx\ⅆy$

$A≔\mathrm{value}\left(q\right)$

$\frac{1}{4}$

Obtain the first moments $M_x$ and $M_y$ 

$q≔\mathrm{Int}\left(y\,\left[x\=0..y-y^3\,y\=0..1\right]\right)$

${\∫}_0^1{\∫}_0^{-y^3\+y}y\ⅆx\ⅆy$

$M_x≔\mathrm{value}\left(q\right)$

$\frac{2}{15}$

$q≔\mathrm{Int}\left(x\,\left[x\=0..y-y^3\,y\=0..1\right]\right)$

${\∫}_0^1{\∫}_0^{-y^3\+y}x\ⅆx\ⅆy$

$M_y≔\mathrm{value}\left(q\right)$

$\frac{4}{105}$

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

$\left[\frac{M_y}{A}\,\frac{M_x}{A}\right]$ = $\left[\frac{16}{105}\,\frac{8}{15}\right]$$$

Table 6.5.9(b)   Coordinates of the centroid calculated from first principles

Obtain the inert integrals defining the center of mass

$\mathrm{Student}:-\mathrm{MultivariateCalculus}:-\mathrm{CenterOfMass}\left(1\,x\=0..y-y^3\,y\=0..1\,\mathrm{output}\=\mathrm{integral}\right)$

$\frac{{\∫}_0^1{\∫}_0^{-y^3\+y}x\ⅆx\ⅆy}{{\∫}_0^1{\∫}_0^{-y^3\+y}1\ⅆx\ⅆy}\,\frac{{\∫}_0^1{\∫}_0^{-y^3\+y}y\ⅆx\ⅆy}{{\∫}_0^1{\∫}_0^{-y^3\+y}1\ⅆx\ⅆy}$

Obtain the coordinates of the center of mass

$\mathrm{Student}:-\mathrm{MultivariateCalculus}:-\mathrm{CenterOfMass}\left(1\,x\=0..y-y^3\,y\=0..1\right)$

$\frac{16}{105}\,\frac{8}{15}$

Obtain a graph of the region $R$, the density $\mathrm{\ρ}\=1$, and the centroid

$\mathrm{Student}:-\mathrm{MultivariateCalculus}:-\mathrm{CenterOfMass}\left(1\,x\=0..y-y^3\,y\=0..1\,\mathrm{output}\=\mathrm{plot}\,\mathrm{caption}\=''''\,\mathrm{scaling}\=\mathrm{constrained}\,\mathrm{labels}\=\left[x\,y\,z\right]\,\mathrm{axes}\=\mathrm{frame}\,\mathrm{orientation}\=\left[-110\,60\,0\right]\,\mathrm{tickmarks}\=\left[2\,2\,2\right]\right)$

Table 6.5.9(c)   Coordinates of the centroid calculated with the CenterOfMass command