$m\={\int }_{-1}^2{\int }_{y^2}^{y\+2}\mathrm{\rho } \mathit{\ⅆ}x \mathit{\ⅆ}y$ = $\frac{6183}{1400}$

$I_x\={\int }_{-1}^2{\int }_{y^2}^{y\+2}\mathrm{\ρ}\cdot y^2 \mathit{\ⅆ}x \mathit{\ⅆ}y$ = $\frac{8451}{1400}$

$I_y\={\int }_{-1}^2{\int }_{y^2}^{y\+2}\mathrm{\ρ}\cdot x^2 \mathit{\ⅆ}x \mathit{\ⅆ}y$ = $\frac{16983}{616}$

$R_x\=\sqrt{\frac{I_y}{m}}\=\sqrt{\frac{16983\/616}{6183\/1400}}$ = $\frac{5}{2519}\sqrt{1584451}$ ≐ 2.50

$R_y\=\sqrt{\frac{I_x}{m}}\=\sqrt{\frac{8451\/1400}{6183\/1400}}$ = $\frac{1}{229}\sqrt{71677}$ ≐ 1.17

Table 6.6.7(a)   Moments of inertial and radii of gyration

Obtain the intersections of the bounding curves

$x\=y^2\,y\=x-2$

$x\=y^2\,y\=x-2$

$\stackrel{\text{solve}}{\to }$

$\left\{x\=1\,y\=-1\right\}\,\left\{x\=4\,y\=2\right\}$

Define the density function $\mathrm{\ρ}\left(x\,y\right)$ 

$\mathrm{\ρ}\=\left(1\+2 x^2\+3 y^2\right)\/10$$\stackrel{\text{assign}}{\to }$

Obtain $m$, the total mass of the lamina

${\int }_{-1}^2{\int }_{y^2}^{y\+2}\mathrm{\ρ} \mathit{\ⅆ}x \mathit{\ⅆ}y$ = $\frac{6183}{1400}$$\stackrel{\text{assign to a name}}{\to }$$m$

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

${\int }_{-1}^2{\int }_{y^2}^{y\+2}\mathrm{\ρ}\cdot y^2 \mathit{\ⅆ}x \mathit{\ⅆ}y$ = $\frac{8451}{1400}$$\stackrel{\text{assign to a name}}{\to }$$\mathrm{Ix}$

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

${\int }_{-1}^2{\int }_{y^2}^{y\+2}\mathrm{\ρ}\cdot x^2 \mathit{\ⅆ}x \mathit{\ⅆ}y$ = $\frac{16983}{616}$$\stackrel{\text{assign to a name}}{\to }$$\mathrm{Iy}$

Obtain $R_x$ 

$\sqrt{\mathrm{Iy}\/m}$ = $\frac{5}{2519}\sqrt{1584451}$$\stackrel{\text{at 5 digits}}{\to }$$2.4986$

Obtain $R_y$ 

$\sqrt{\mathrm{Ix}\/m}$ = $\frac{1}{229}\sqrt{71677}$$\stackrel{\text{at 5 digits}}{\to }$$1.1691$

Table 6.6.7(b)   Moments of inertia and radii of gyration from first principles

Initialize

$\mathrm{\ρ}≔\left(1\+2 x^2\+ 3 y^2\right)\/10\:$

Obtain the total mass of the lamina

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

${\∫}_{-1}^2{\∫}_{y^2}^{y\+2}\left(\frac{1}{5}{x}^2\+\frac{3}{10}{y}^2\+\frac{1}{10}\right)\ⅆx\ⅆy$

$m≔\mathrm{value}\left(q\right)$

$\frac{6183}{1400}$

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

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

${\∫}_{-1}^2{\∫}_{y^2}^{y\+2}\left(\frac{1}{5}{x}^2\+\frac{3}{10}{y}^2\+\frac{1}{10}\right)y^2\ⅆx\ⅆy$

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

$\frac{8451}{1400}$

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

${\∫}_{-1}^2{\∫}_{y^2}^{y\+2}\left(\frac{1}{5}{x}^2\+\frac{3}{10}{y}^2\+\frac{1}{10}\right)x^2\ⅆx\ⅆy$

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

$\frac{16983}{616}$

Obtain the radii of gyration

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

$R_x\=\frac{5}{2519}\sqrt{1584451}$

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

$R_y\=\frac{1}{229}\sqrt{71677}$

Table 6.6.7(c)   Moments of inertia and radii of gyration from first principles