Initialize

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

$\mathrm{Ix}\={\int }_0^{2 \mathrm{\pi }}{\int }_0^2{\int }_0^{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{40534016}{175175}$$\stackrel{\text{at 5 digits}}{\to }$$231.39$$$

$\mathrm{Iy}\={\int }_0^{2 \mathrm{\pi }}{\int }_0^2{\int }_0^{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{40321024}{175175}$$\stackrel{\text{at 5 digits}}{\to }$$230.18$$$

$\mathrm{Iz}\={\int }_0^{2 \mathrm{\pi }}{\int }_0^2{\int }_0^{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{16384}{385}$$\stackrel{\text{at 5 digits}}{\to }$$42.556$$$

Table 8.4.11(a)   Calculations for the moments of inertia

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

$m$ = $\frac{4096}{105}$$$

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

$k_x$ = $\frac{2}{5005}\sqrt{37147110}$$\stackrel{\text{at 5 digits}}{\to }$$2.4355$$$

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

$k_y$ = $\frac{2}{5005}\sqrt{36951915}$$\stackrel{\text{at 5 digits}}{\to }$$2.4291$$$

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

$k_z$ = $\frac{2}{11}\sqrt{33}$$\stackrel{\text{at 5 digits}}{\to }$$1.0445$$$

Table 8.4.11(b)   Radii of gyration

Initialize

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

Obtain the moments of inertia

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

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

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

$\frac{40534016}{175175}$

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

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

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

$\frac{40321024}{175175}$

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

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

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

$\frac{16384}{385}$

Obtain the total mass $m$ 

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

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

$m≔\mathrm{value}\left(M\right)$

$\frac{4096}{105}$

Obtain the radii of gyration