Initialize

$\mathrm{\δ}\=3\+x\+y\+z$$\stackrel{\text{assign}}{\to }$

$\mathrm{Ix}\={\int }_{-1}^1{\int }_{-\sqrt{1-x^2}}^{{\sqrt{1-x^2}}_{}}{\int }_{-\sqrt{1-x^2}}^{{\sqrt{1-x^2}}_{}}\mathrm{\δ} \left(y^2\+z^2\right) \mathit{\ⅆ}z \mathit{\ⅆ}y \mathit{\ⅆ}x$$\stackrel{\text{assign}}{\to }$

$\mathrm{Ix}$ = $\frac{128}{15}$$\stackrel{\text{at 5 digits}}{\to }$$8.5333$$$

$\mathrm{Iy}\={\int }_{-1}^1{\int }_{-\sqrt{1-x^2}}^{{\sqrt{1-x^2}}_{}}{\int }_{-\sqrt{1-x^2}}^{{\sqrt{1-x^2}}_{}}\mathrm{\delta } \left(x^2\+z^2\right) \mathit{\ⅆ}z \mathit{\ⅆ}y \mathit{\ⅆ}x$$\stackrel{\text{assign}}{\to }$

$\mathrm{Iy}$ = $\frac{112}{15}$$\stackrel{\text{at 5 digits}}{\to }$$7.4667$$$

$\mathrm{Iz}\={\int }_{-1}^1{\int }_{-\sqrt{1-x^2}}^{{\sqrt{1-x^2}}_{}}{\int }_{-\sqrt{1-x^2}}^{{\sqrt{1-x^2}}_{}}\mathrm{\delta } \left(x^2\+y^2\right) \mathit{\ⅆ}z \mathit{\ⅆ}y \mathit{\ⅆ}x$$\stackrel{\text{assign}}{\to }$

$\mathrm{Iz}$ = $\frac{112}{15}$$\stackrel{\text{at 5 digits}}{\to }$$7.4667$$$

Table 8.4.3(a)   Calculations for the moments of inertia

$m\={\int }_{-1}^1{\int }_{-\sqrt{1-x^2}}^{{\sqrt{1-x^2}}_{}}{\int }_{-\sqrt{1-x^2}}^{{\sqrt{1-x^2}}_{}}\mathrm{\δ} \mathit{\ⅆ}z \mathit{\ⅆ}y \mathit{\ⅆ}x$$\stackrel{\text{assign}}{\to }$

$m$ = $16$$$

$k_x\=\sqrt{\mathrm{Ix}\/m}$$\stackrel{\text{assign}}{\to }$

$k_x$ = $\frac{2}{15}\sqrt{30}$$\stackrel{\text{at 5 digits}}{\to }$$0.73028$$$

$k_y\=\sqrt{\mathrm{Iy}\/m}$$\stackrel{\text{assign}}{\to }$

$k_y$ = $\frac{1}{15}\sqrt{105}$$\stackrel{\text{at 5 digits}}{\to }$$0.68314$$$

$k_z\=\sqrt{\mathrm{Iz}\/m}$$\stackrel{\text{assign}}{\to }$

$k_z$ = $\frac{1}{15}\sqrt{105}$$\stackrel{\text{at 5 digits}}{\to }$$0.68314$$$

Table 8.4.3(b)   Radii of gyration

Initialize

$\mathrm{\δ}≔3\+x\+y\+z\:$

Obtain the moments of inertia

$Q_x≔\mathrm{Int}\left(\mathrm{\δ} \left(y^2\+z^2\right)\,\left[z\=-\sqrt{1-x^2}..\sqrt{1-x^2}\,y\=-\sqrt{1-x^2}..\sqrt{1-x^2}\,x\=-1..1\right]\right)$

${\∫}_{-1}^1{\∫}_{-\sqrt{-x^2\+1}}^{\sqrt{-x^2\+1}}{\∫}_{-\sqrt{-x^2\+1}}^{\sqrt{-x^2\+1}}\left(3\+x\+y\+z\right)\left(y^2\+z^2\right)\ⅆz\ⅆy\ⅆx$

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

$\frac{128}{15}$

$Q_y≔\mathrm{Int}\left(\mathrm{\delta } \left(x^2\+z^2\right)\,\left[z\=-\sqrt{1-x^2}..\sqrt{1-x^2}\,y\=-\sqrt{1-x^2}..\sqrt{1-x^2}\,x\=-1..1\right]\right)$

${\∫}_{-1}^1{\∫}_{-\sqrt{-x^2\+1}}^{\sqrt{-x^2\+1}}{\∫}_{-\sqrt{-x^2\+1}}^{\sqrt{-x^2\+1}}\left(3\+x\+y\+z\right)\left(x^2\+z^2\right)\ⅆz\ⅆy\ⅆx$

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

$\frac{112}{15}$

$Q_z≔\mathrm{Int}\left(\mathrm{\delta } \left(x^2\+y^2\right)\,\left[z\=-\sqrt{1-x^2}..\sqrt{1-x^2}\,y\=-\sqrt{1-x^2}..\sqrt{1-x^2}\,x\=-1..1\right]\right)$

${\∫}_{-1}^1{\∫}_{-\sqrt{-x^2\+1}}^{\sqrt{-x^2\+1}}{\∫}_{-\sqrt{-x^2\+1}}^{\sqrt{-x^2\+1}}\left(3\+x\+y\+z\right)\left(x^2\+y^2\right)\ⅆz\ⅆy\ⅆx$

$\mathrm{Iz}≔\mathrm{value}\left(Q_z\right)$

$\frac{112}{15}$

Obtain the total mass $m$ 

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

${\∫}_{-1}^1{\∫}_{-\sqrt{-x^2\+1}}^{\sqrt{-x^2\+1}}{\∫}_{-\sqrt{-x^2\+1}}^{\sqrt{-x^2\+1}}\left(3\+x\+y\+z\right)\ⅆz\ⅆy\ⅆx$

$m≔\mathrm{value}\left(M\right)$

$16$

Obtain the radii of gyration