Initialize

$\mathrm{\δ}\=2 x^2\+3 y^2\+4 z^2$$\stackrel{\text{assign}}{\to }$

$\mathrm{Ix}\={\int }_0^2{\int }_0^4{\int }_0^{\sqrt{4-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{12656}{15}\mathrm{\π}$$\stackrel{\text{at 5 digits}}{\to }$$2650.7$$$

$\mathrm{Iy}\={\int }_0^2{\int }_0^4{\int }_0^{\sqrt{4-x^2}}\mathrm{\δ} \left(x^2\+z^2\right) \mathit{\ⅆ}z \mathit{\ⅆ}y \mathit{\ⅆ}x$$\stackrel{\text{assign}}{\to }$$$$$

$\mathrm{Iy}$ = $192\mathrm{\π}$$\stackrel{\text{at 5 digits}}{\to }$$603.19$$$

$\mathrm{Iz}\={\int }_0^2{\int }_0^4{\int }_0^{\sqrt{4-x^2}}\mathrm{\δ} \left(x^2\+y^2\right) \mathit{\ⅆ}z \mathit{\ⅆ}y \mathit{\ⅆ}x$$\stackrel{\text{assign}}{\to }$$$

$\mathrm{Iz}$ = $\frac{12496}{15}\mathrm{\π}$$\stackrel{\text{at 5 digits}}{\to }$$2617.2$$$

Table 8.4.10(a)   Calculations for the moments of inertia

$m\={\int }_0^2{\int }_0^4{\int }_0^{\sqrt{4-x^2}}\mathrm{\δ} \mathit{\ⅆ}z \mathit{\ⅆ}y \mathit{\ⅆ}x$$\stackrel{\text{assign}}{\to }$

$m$ = $88\mathrm{\π}$$$

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

$k_x$ = $\frac{1}{165}\sqrt{261030}$$\stackrel{\text{at 5 digits}}{\to }$$3.0964$$$

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

$k_y$ = $\frac{2}{11}\sqrt{66}$$\stackrel{\text{at 5 digits}}{\to }$$1.4771$$$

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

$k_z$ = $\frac{1}{15}\sqrt{2130}$$\stackrel{\text{at 5 digits}}{\to }$$3.0768$$$

Table 8.4.10(b)   Radii of gyration

Initialize

$\mathrm{\δ}≔2 x^2\+3 y^2\+4 z^2\:$

Obtain the moments of inertia

$Q_x≔\mathrm{Int}\left(\mathrm{\δ} \left(y^2\+z^2\right)\,\left[z\=0..\sqrt{4-x^2}\,y\=0..4\,x\=0..2\right]\right)$

${\∫}_0^2{\∫}_0^4{\∫}_0^{\sqrt{-x^2\+4}}\left(2x^2\+3y^2\+4z^2\right)\left(y^2\+z^2\right)\ⅆz\ⅆy\ⅆx$

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

$\frac{12656}{15}\mathrm{\π}$

$Q_y≔\mathrm{Int}\left(\mathrm{\delta } \left(x^2\+z^2\right)\,\left[z\=0..\sqrt{4-x^2}\,y\=0..4\,x\=0..2\right]\right)$

${\∫}_0^2{\∫}_0^4{\∫}_0^{\sqrt{-x^2\+4}}\left(2x^2\+3y^2\+4z^2\right)\left(x^2\+z^2\right)\ⅆz\ⅆy\ⅆx$

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

$192\mathrm{\π}$

$Q_z≔\mathrm{Int}\left(\mathrm{\delta } \left(x^2\+y^2\right)\,\left[z\=0..\sqrt{4-x^2}\,y\=0..4\,x\=0..2\right]\right)$

${\∫}_0^2{\∫}_0^4{\∫}_0^{\sqrt{-x^2\+4}}\left(2x^2\+3y^2\+4z^2\right)\left(x^2\+y^2\right)\ⅆz\ⅆy\ⅆx$

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

$\frac{12496}{15}\mathrm{\π}$

Obtain the total mass $m$ 

$M≔\mathrm{Int}\left(\mathrm{\δ}\,\left[z\=0..\sqrt{4-x^2}\,y\=0..4\,x\=0..2\right]\right)$

${\∫}_0^2{\∫}_0^4{\∫}_0^{\sqrt{-x^2\+4}}\left(2x^2\+3y^2\+4z^2\right)\ⅆz\ⅆy\ⅆx$

$m≔\mathrm{value}\left(M\right)$

$88\mathrm{\π}$

Obtain the radii of gyration