Initialize

$\mathrm{\δ}\=\left(r^2\+z\right) \sin \left(\mathrm{\theta }\/4\right)$$\stackrel{\text{assign}}{\to }$

$\mathrm{Ix}\={\int }_0^{2 \mathrm{\pi }}{\int }_0^3{\int }_0^{5-r \sin \left(\mathrm{\theta }\right)}\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{183193671}{13090}$$\stackrel{\text{at 5 digits}}{\to }$$13995.$$$

$\mathrm{Iy}\={\int }_0^{2 \mathrm{\pi }}{\int }_0^3{\int }_0^{5-r \sin \left(\mathrm{\theta }\right)}\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{232704750}{17017}$$\stackrel{\text{at 5 digits}}{\to }$$13675.$$$

$\mathrm{Iz}\={\int }_0^{2 \mathrm{\pi }}{\int }_0^3{\int }_0^{5-r \sin \left(\mathrm{\theta }\right)}\mathrm{\delta } \left(r^2\right) r \mathit{\ⅆ}z \mathit{\ⅆ}r \mathit{\ⅆ}\mathrm{\θ}$$\stackrel{\text{assign}}{\to }$$$

$\mathrm{Iz}$ = $\frac{291087}{70}$$\stackrel{\text{at 5 digits}}{\to }$$4158.4$$$

Table 8.4.5(a)   Calculations for the moments of inertia

$m\={\int }_0^{2 \mathrm{\π}}{\int }_0^3{\int }_0^{5- r \sin \left(\mathrm{\theta }\right)}\mathrm{\δ} r \mathit{\ⅆ}z \mathit{\ⅆ}r \mathit{\ⅆ}\mathrm{\θ}$$\stackrel{\text{assign}}{\to }$$$

$m$ = $\frac{131322}{175}$$$

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

$k_x$ = $\frac{1}{8185738}\sqrt{1249646161720165}$$\stackrel{\text{at 5 digits}}{\to }$$4.3184$$$

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

$k_y$ = $\frac{25}{53207297}\sqrt{82543938310405}$$\stackrel{\text{at 5 digits}}{\to }$$4.2689$$$

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

$k_z$ = $\frac{3}{43774}\sqrt{1179818735}$$\stackrel{\text{at 5 digits}}{\to }$$2.3540$$$

Table 8.4.5(b)   Radii of gyration

Initialize

$\mathrm{\δ}≔\left(r^2\+z\right) \sin \left(\mathrm{\theta }\/4\right)\:$

Obtain the moments of inertia

$Q_x≔\mathrm{Int}\left(r \mathrm{\δ} \left(r^2{\sin }^2\left(\mathrm{\θ}\right)\+z^2\right)\,\left[z\=0..5-r \sin \left(\mathrm{\theta }\right)\,r\=0..3\,\mathrm{\θ}\=0..2 \mathrm{\π}\right]\right)$

${\∫}_0^{2\mathrm{\π}}{\∫}_0^3{\∫}_0^{5-r\sin \left(\mathrm{\θ}\right)}r\left(r^2\+z\right)\sin \left(\frac{1}{4}\mathrm{\θ}\right)\left(r^2{\sin \left(\mathrm{\θ}\right)}^2\+z^2\right)\ⅆz\ⅆr\ⅆ\mathrm{\θ}$

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

$\frac{183193671}{13090}$

$Q_y≔\mathrm{Int}\left(r \mathrm{\δ} \left(r^2{\cos }^2\left(\mathrm{\θ}\right)\+z^2\right)\,\left[z\=0..5-r \sin \left(\mathrm{\theta }\right)\,r\=0..3\,\mathrm{\θ}\=0..2 \mathrm{\π}\right]\right)$

${\∫}_0^{2\mathrm{\π}}{\∫}_0^3{\∫}_0^{5-r\sin \left(\mathrm{\θ}\right)}r\left(r^2\+z\right)\sin \left(\frac{1}{4}\mathrm{\θ}\right)\left(r^2{\cos \left(\mathrm{\θ}\right)}^2\+z^2\right)\ⅆz\ⅆr\ⅆ\mathrm{\θ}$

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

$\frac{232704750}{17017}$

$Q_z≔\mathrm{Int}\left(r \mathrm{\δ} \left(r^2\right)\,\left[z\=0..5-r \sin \left(\mathrm{\theta }\right)\,r\=0..3\,\mathrm{\θ}\=0..2 \mathrm{\π}\right]\right)$

${\∫}_0^{2\mathrm{\π}}{\∫}_0^3{\∫}_0^{5-r\sin \left(\mathrm{\θ}\right)}{r}^3\left(r^2\+z\right)\sin \left(\frac{1}{4}\mathrm{\θ}\right)\ⅆz\ⅆr\ⅆ\mathrm{\θ}$

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

$\frac{291087}{70}$

Obtain the total mass $m$ 

$M≔\mathrm{Int}\left(r \mathrm{\δ}\,\left[z\=0..5-r \sin \left(\mathrm{\theta }\right)\,r\=0..3\,\mathrm{\θ}\=0..2 \mathrm{\π}\right]\right)$

${\∫}_0^{2\mathrm{\π}}{\∫}_0^3{\∫}_0^{5-r\sin \left(\mathrm{\θ}\right)}r\left(r^2\+z\right)\sin \left(\frac{1}{4}\mathrm{\θ}\right)\ⅆz\ⅆr\ⅆ\mathrm{\θ}$

$m≔\mathrm{value}\left(M\right)$

$\frac{131322}{175}$

Obtain the radii of gyration