Initialize

$\mathrm{\δ}\=x y^2{z}^3$$\stackrel{\text{assign}}{\to }$

$\mathrm{Ix}\={\int }_0^1{\int }_0^{2-x}{\int }_0^{2-x-y}\mathrm{\δ} \left(y^2\+z^2\right) \mathit{\ⅆ}z \mathit{\ⅆ}y \mathit{\ⅆ}x$$\stackrel{\text{assign}}{\to }$

$\mathrm{Ix}$ = $\frac{1018}{51975}$$\stackrel{\text{at 5 digits}}{\to }$$0.019586$$$

$\mathrm{Iy}\={\int }_0^1{\int }_0^{2-x}{\int }_0^{2-x-y}\mathrm{\delta } \left(x^2\+z^2\right) \mathit{\ⅆ}z \mathit{\ⅆ}y \mathit{\ⅆ}x$$\stackrel{\text{assign}}{\to }$

$\mathrm{Iy}$ = $\frac{1613}{103950}$$\stackrel{\text{at 5 digits}}{\to }$$0.015517$$$

$\mathrm{Iz}\={\int }_0^1{\int }_0^{2-x}{\int }_0^{2-x-y}\mathrm{\delta } \left(x^2\+y^2\right) \mathit{\ⅆ}z \mathit{\ⅆ}y \mathit{\ⅆ}x$$\stackrel{\text{assign}}{\to }$

$\mathrm{Iz}$ = $\frac{184}{17325}$$\stackrel{\text{at 5 digits}}{\to }$$0.010620$$$

Table 8.4.12(a)   Calculations for the moments of inertia

$m\={\int }_0^1{\int }_0^{2-x}{\int }_0^{2-x-y}\mathrm{\δ} \mathit{\ⅆ}z \mathit{\ⅆ}y \mathit{\ⅆ}x$$\stackrel{\text{assign}}{\to }$

$m$ = $\frac{251}{15120}$$$

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

$k_x$ = $\frac{4}{13805}\sqrt{14053490}$$\stackrel{\text{at 5 digits}}{\to }$$1.0862$$$

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

$k_y$ = $\frac{2}{13805}\sqrt{44534930}$$\stackrel{\text{at 5 digits}}{\to }$$0.96686$$$

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

$k_z$ = $\frac{8}{13805}\sqrt{1905090}$$\stackrel{\text{at 5 digits}}{\to }$$0.79988$$$

Table 8.4.12(b)   Radii of gyration

Initialize

$\mathrm{\δ}≔x y^2{z}^3\:$

Obtain the moments of inertia

$Q_x≔\mathrm{Int}\left(\mathrm{\δ} \left(y^2\+z^2\right)\,\left[z\=0..2-x-y\,y\=0..2-x\,x\=0..1\right]\right)$

${\∫}_0^1{\∫}_0^{2-x}{\∫}_0^{2-x-y}x{y}^2{z}^3\left(y^2\+z^2\right)\ⅆz\ⅆy\ⅆx$

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

$\frac{1018}{51975}$

$Q_y≔\mathrm{Int}\left(\mathrm{\delta } \left(x^2\+z^2\right)\,\left[z\=0..2-x-y\,y\=0..2-x\,x\=0..1\right]\right)$

${\∫}_0^1{\∫}_0^{2-x}{\∫}_0^{2-x-y}x{y}^2{z}^3\left(x^2\+z^2\right)\ⅆz\ⅆy\ⅆx$

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

$\frac{1613}{103950}$

$Q_z≔\mathrm{Int}\left(\mathrm{\delta } \left(x^2\+y^2\right)\,\left[z\=0..2-x-y\,y\=0..2-x\,x\=0..1\right]\right)$

${\∫}_0^1{\∫}_0^{2-x}{\∫}_0^{2-x-y}x{y}^2{z}^3\left(x^2\+y^2\right)\ⅆz\ⅆy\ⅆx$

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

$\frac{184}{17325}$

Obtain the total mass $m$ 

$M≔\mathrm{Int}\left(\mathrm{\δ}\,\left[z\=0..2-x-y\,y\=0..2-x\,x\=0..1\right]\right)$

${\∫}_0^1{\∫}_0^{2-x}{\∫}_0^{2-x-y}x{y}^2{z}^3\ⅆz\ⅆy\ⅆx$

$m≔\mathrm{value}\left(M\right)$

$\frac{251}{15120}$

Obtain the radii of gyration