Initialize

$\mathrm{\δ}\=r z \cos \left(\mathrm{\theta }\/6\right)$$\stackrel{\text{assign}}{\to }$

$\mathrm{Ix}\={\int }_0^{2 \mathrm{\pi }}{\int }_0^1{\int }_{3 r^2}^{4-r^2}\mathrm{\δ} \left(r^2{\sin }^2\left(\mathrm{\θ}\right)\+z^2\right) r \mathit{\ⅆ}z \mathit{\ⅆ}r \mathit{\ⅆ}\mathrm{\θ}$$\stackrel{\text{assign}}{\to }$

$\mathrm{Ix}$ = $\frac{450152}{15015}\sqrt{3}$$\stackrel{\text{at 5 digits}}{\to }$$51.928$

$\mathrm{Iy}\={\int }_0^{2 \mathrm{\pi }}{\int }_0^1{\int }_{3 r^2}^{4-r^2}\mathrm{\delta } \left(r^2{\cos }^2\left(\mathrm{\theta }\right)\+z^2\right) r \mathit{\ⅆ}z \mathit{\ⅆ}r \mathit{\ⅆ}\mathrm{\θ}$$\stackrel{\text{assign}}{\to }$

$\mathrm{Iy}$ = $\frac{449968}{15015}\sqrt{3}$$\stackrel{\text{at 5 digits}}{\to }$$51.908$

$\mathrm{Iz}\={\int }_0^{2 \mathrm{\pi }}{\int }_0^1{\int }_{3 r^2}^{4-r^2}\mathrm{\delta } \left(r^2\right) r \mathit{\ⅆ}z \mathit{\ⅆ}r \mathit{\ⅆ}\mathrm{\θ}$$\stackrel{\text{assign}}{\to }$

$\mathrm{Iz}$ = $\frac{184}{105}\sqrt{3}$$\stackrel{\text{at 5 digits}}{\to }$$3.0353$

Table 8.4.8(a)   Calculations for the moments of inertia

$m\={\int }_0^{2 \mathrm{\pi }}{\int }_0^1{\int }_{3 r^2}^{4- r^2}\mathrm{\δ} r \mathit{\ⅆ}z \mathit{\ⅆ}r \mathit{\ⅆ}\mathrm{\θ}$$\stackrel{\text{assign}}{\to }$

$m$ = $\frac{136}{35}\sqrt{3}$

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

$k_x$ = $\frac{1}{7293}\sqrt{410369817}$$\stackrel{\text{at 5 digits}}{\to }$$2.7778$

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

$k_y$ = $\frac{1}{7293}\sqrt{410202078}$$\stackrel{\text{at 5 digits}}{\to }$$2.7771$

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

$k_z$ = $\frac{1}{51}\sqrt{1173}$$\stackrel{\text{at 5 digits}}{\to }$$0.67155$

Table 8.4.8(b)   Radii of gyration

Initialize

$\mathrm{\δ}≔r z \cos \left(\mathrm{\theta }\/6\right)\:$

Obtain the moments of inertia

$Q_x≔\mathrm{Int}\left(r \mathrm{\δ} \left(r^2{\sin }^2\left(\mathrm{\θ}\right)\+z^2\right)\,\left[z\=3 r^2..4- r^2\,r\=0..1\,\mathrm{\θ}\=0..2 \mathrm{\π}\right]\right)$

${\∫}_0^{2\mathrm{\π}}{\∫}_0^1{\∫}_{3r^2}^{-r^2\+4}{r}^{2}z\cos \left(\frac{1}{6}\mathrm{\θ}\right)\left(r^2{\sin \left(\mathrm{\θ}\right)}^2\+z^2\right)\ⅆz\ⅆr\ⅆ\mathrm{\θ}$

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

$\frac{450152}{15015}\sqrt{3}$

$Q_y≔\mathrm{Int}\left(r \mathrm{\δ} \left(r^2{\cos }^2\left(\mathrm{\θ}\right)\+z^2\right)\,\left[z\=3 r^2..4- r^2\,r\=0..1\,\mathrm{\θ}\=0..2 \mathrm{\π}\right]\right)$

${\∫}_0^{2\mathrm{\π}}{\∫}_0^1{\∫}_{3r^2}^{-r^2\+4}{r}^{2}z\cos \left(\frac{1}{6}\mathrm{\θ}\right)\left(r^2{\cos \left(\mathrm{\θ}\right)}^2\+z^2\right)\ⅆz\ⅆr\ⅆ\mathrm{\θ}$

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

$\frac{449968}{15015}\sqrt{3}$

$Q_z≔\mathrm{Int}\left(r \mathrm{\δ} \left(r^2\right)\,\left[z\=3 r^2..4- r^2\,r\=0..1\,\mathrm{\θ}\=0..2 \mathrm{\π}\right]\right)$

${\∫}_0^{2\mathrm{\π}}{\∫}_0^1{\∫}_{3r^2}^{-r^2\+4}{r}^{4}z\cos \left(\frac{1}{6}\mathrm{\θ}\right)\ⅆz\ⅆr\ⅆ\mathrm{\θ}$

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

$\frac{184}{105}\sqrt{3}$

Obtain the total mass $m$ 

$M≔\mathrm{Int}\left(r \mathrm{\δ}\,\left[z\=3 r^2..4- r^2\,r\=0..1\,\mathrm{\θ}\=0..2 \mathrm{\π}\right]\right)$

${\∫}_0^{2\mathrm{\π}}{\∫}_0^1{\∫}_{3r^2}^{-r^2\+4}{r}^{2}z\cos \left(\frac{1}{6}\mathrm{\θ}\right)\ⅆz\ⅆr\ⅆ\mathrm{\θ}$

$m≔\mathrm{value}\left(M\right)$

$\frac{136}{35}\sqrt{3}$

Obtain the radii of gyration