Initialize

$\mathrm{\δ}\=r z^2\cos \left(\mathrm{\theta }\/3\right)$$\stackrel{\text{assign}}{\to }$

$\mathrm{Ix}\={\int }_0^{\mathrm{\pi }}{\int }_0^2{\int }_0^{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{2187}{140}$$\stackrel{\text{at 5 digits}}{\to }$$15.621$$$

$\mathrm{Iy}\={\int }_0^{\mathrm{\pi }}{\int }_0^2{\int }_0^{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{4617}{560}$$\stackrel{\text{at 5 digits}}{\to }$$8.2446$$$

$\mathrm{Iz}\={\int }_0^{\mathrm{\pi }}{\int }_0^2{\int }_0^{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{243}{20}$$\stackrel{\text{at 5 digits}}{\to }$$12.150$$$

Table 8.4.2(a)   Calculations for the moments of inertia

$m\={\int }_0^{\mathrm{\pi }}{\int }_0^2{\int }_0^{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{81}{20}$$$

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

$k_x$ = $\frac{3}{7}\sqrt{21}$$\stackrel{\text{at 5 digits}}{\to }$$1.9640$$$

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

$k_y$ = $\frac{1}{14}\sqrt{399}$$\stackrel{\text{at 5 digits}}{\to }$$1.4268$$$

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

$k_z$ = $\sqrt{3}$$\stackrel{\text{at 5 digits}}{\to }$$1.7321$$$

Table 8.4.2(b)   Radii of gyration

Initialize

$\mathrm{\δ}≔r z^2\cos \left(\mathrm{\theta }\/3\right)\:$

Obtain the moments of inertia

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

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

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

$\frac{2187}{140}$

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

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

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

$\frac{4617}{560}$

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

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

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

$\frac{243}{20}$

Obtain the total mass $m$ 

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

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

$m≔\mathrm{value}\left(M\right)$

$\frac{81}{20}$

Obtain the radii of gyration