$m\={\int }_0^1{\int }_0^{y-y^3}\mathit{\ⅆ}x \mathit{\ⅆ}y$ = $\frac{1}{4}$

$I_x\={\int }_0^1{\int }_0^{y-y^3}{y}^2 \mathit{\ⅆ}x \mathit{\ⅆ}y$ = $\frac{1}{12}$

$I_y\={\int }_0^1{\int }_0^{y-y^3}{x}^2 \mathit{\ⅆ}x \mathit{\ⅆ}y$ = $\frac{1}{120}$

$R_x\=\sqrt{I_y\/m}\=\sqrt{\frac{1\/120}{1\/4}}\=\frac{1}{\sqrt{30}}$ ≐ 0.58

$R_y\=\sqrt{\mathrm{Ix}\/m}\=\sqrt{\frac{1\/12}{1\/4}}\=\frac{1}{\sqrt{3}}$ ≐ 0.18

Table 6.6.9(a)   Moments of inertia and radii of gyration

Obtain $m$, the total mass of the lamina

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

Obtain $I_x$, the moment of inertia about the $x$-axis

${\int }_0^1{\int }_0^{y-y^3}{y}^2 \mathit{\ⅆ}x \mathit{\ⅆ}y$ = $\frac{1}{12}$$\stackrel{\text{assign to a name}}{\to }$$\mathrm{Ix}$ 

Obtain $I_y$, the moment of inertia about the $y$-axis

${\int }_0^1{\int }_0^{y-y^3}{x}^2 \mathit{\ⅆ}x \mathit{\ⅆ}y$ = $\frac{1}{120}$$\stackrel{\text{assign to a name}}{\to }$$\mathrm{Iy}$

Obtain $R_x$ 

$\sqrt{\mathrm{Iy}\/m}$ = $\frac{1}{30}\sqrt{30}$$\stackrel{\text{at 10 digits}}{\to }$$0.1825741858$

Obtain $R_y$ 

$\sqrt{\mathrm{Ix}\/m}$ = $\frac{1}{3}\sqrt{3}$$\stackrel{\text{at 10 digits}}{\to }$$0.5773502693$

Table 6.6.9(b)   Moments of inertia and radii of gyration

Obtain the total mass of the lamina

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

${\∫}_0^1{\∫}_0^{-y^3\+y}1\ⅆx\ⅆy$

$m≔\mathrm{value}\left(q\right)$

$\frac{1}{4}$

Obtain the moments of inertia $I_x$ and $I_y$ 

$q≔\mathrm{Int}\left(y^2\,\left[x\=0..y-y^3\,y\=0..1\right]\right)$

${\∫}_0^1{\∫}_0^{-y^3\+y}{y}^2\ⅆx\ⅆy$

$\mathrm{Ix}≔\mathrm{value}\left(q\right)$

$\frac{1}{12}$

$q≔\mathrm{Int}\left(x^2\,\left[x\=0..y-y^3\,y\=0..1\right]\right)$

${\∫}_0^1{\∫}_0^{-y^3\+y}{x}^2\ⅆx\ⅆy$

$\mathrm{Iy}≔\mathrm{value}\left(q\right)$

$\frac{1}{120}$

Obtain the radii of gyration

$$\begin{gathered} R_x\=\mathrm{sqrt}\left(\mathrm{Iy}\/m\right)\; \\ {R}_x\=\mathrm{evalf}\left(\mathrm{sqrt}\left(\mathrm{Ix}\/m\right)\right) \end{gathered}$$

$R_x\=\frac{1}{30}\sqrt{30}$

$$\begin{gathered} R_y\=\mathrm{sqrt}\left(\mathrm{Ix}\/m\right)\; \\ {R}_y\=\mathrm{evalf}\left(\mathrm{sqrt}\left(\mathrm{Iy}\/m\right)\right) \end{gathered}$$

$R_y\=\frac{1}{3}\sqrt{3}$

Table 6.6.9(c)   Moments of inertia and radii of gyration