dm = γ * dV = γ * dz * π * 𝜚2
$$\mathrm{I}_{\left( \mathrm{\text{XY}} \right)}\mathrm{=}\int_{\mathrm{m}}^{}{\mathrm{z}^{\mathrm{2}}\mathrm{dm =}\int_{\mathrm{0}}^{\mathrm{h}}{\mathrm{z}^{\mathrm{2}}\mathrm{\text{γπ}}\mathrm{\varrho}^{\mathrm{2}}\mathrm{dz =}}\int_{\mathrm{0}}^{\mathrm{h}}{\mathrm{z}^{\mathrm{2}}\mathrm{\text{γπ}}\left( \mathrm{z*}\frac{\mathrm{r}}{\mathrm{h}} \right)^{\mathrm{2}}\mathrm{dz =}}\mathrm{\text{γπ}}\frac{\mathrm{r}^{\mathrm{2}}}{\mathrm{h}^{\mathrm{2}}}\int_{\mathrm{0}}^{\mathrm{h}}{\mathrm{z}^{\mathrm{4}}\mathrm{dz =}}}$$
$$\mathrm{= \gamma\pi}\frac{\mathrm{r}^{\mathrm{2}}}{\mathrm{h}^{\mathrm{2}}}\frac{\mathrm{z}^{\mathrm{5}}}{\mathrm{5}}\left\{ \begin{matrix}
\mathrm{h} \\
\mathrm{0} \\
\end{matrix} \right.\ \mathrm{= \gamma\pi}\mathrm{r}^{\mathrm{2}}\frac{\mathrm{h}^{\mathrm{5}}}{\mathrm{5}}\mathrm{=}\frac{\mathrm{1}}{\mathrm{3}}\mathrm{\text{γπ}}\mathrm{r}^{\mathrm{2}}\mathrm{h*}\mathrm{h}^{\mathrm{2}}\frac{\mathrm{3}}{\mathrm{5}}\mathrm{=}\frac{\mathrm{3}}{\mathrm{5}}\mathrm{m}\mathrm{h}^{\mathrm{2}}$$
$$m = \gamma dV = \gamma 2\pi\varrho\left( h - \frac{h}{r}\varrho \right)d\varrho\ (rys.2.1)$$
$$I_{z} = \int_{m}^{}\varrho^{2}dm = \int_{0}^{r}\varrho^{2}\gamma 2\pi\varrho\left( h - \frac{h}{r}\varrho \right)\mathrm{d\varrho =}\ \gamma 2\pi h\int_{0}^{r}{\varrho^{3}\text{dϱ}} -$$
$$- \gamma 2\pi\frac{h}{r}\int_{0}^{r}{\varrho^{4}\text{dϱ}} = \ \gamma 2\pi\frac{\varrho^{4}}{4}\left\{ \begin{matrix}
\mathrm{r} \\
\mathrm{0} \\
\end{matrix} \right.\ - \gamma 2\pi\frac{\varrho^{5}}{5r}\left\{ \begin{matrix}
r \\
\mathrm{0} \\
\end{matrix} \right.\ = \gamma 2\pi\frac{r^{4}}{4} - \gamma 2\pi\frac{r^{5}}{5r} =$$
$$\frac{1}{2}\text{πγh}r^{4} - \frac{2}{5}\text{πγh}r^{4} = \frac{1}{10}\text{πγh}r^{4} = \frac{1}{3}\text{πγh}r^{2}*\frac{3}{10}r^{2} = \frac{3}{10}mr^{2}$$
I(YZ) + I(ZX) = IZ
$$I_{\left( \text{YZ} \right)} = I_{\left( \text{ZX} \right)} = \frac{1}{2}I_{Z} = \frac{1}{2}*\frac{3}{10}mr^{2} = \frac{3}{20}mr^{2}$$
$$I_{X} = I_{Y} = \frac{3}{5}mh^{2} + \frac{3}{20}mr^{2}$$
$$\text{Iz}c^{'} \bigtriangleup = \frac{3}{10}mr^{2} = \frac{1}{10}\text{γπ}r^{4}h$$
$$\text{Iz}c^{'} = \frac{1}{2}mr^{2} = \frac{1}{2}\text{γπ}r^{4}h$$
Iz △ =Izc′ △ +m △ r2
Iz = Izc′ + mr2
$$Iz = Iz \bigtriangleup + Iz\ = \frac{1}{10}\text{γπ}r^{4}h + \frac{1}{3}\text{γπ\ }r^{4}h + \frac{1}{2}\text{γπ}r^{4}h + \gamma\pi r^{4}h =$$
$$Iz = \frac{58}{30}\text{γπ}r^{4}h$$