속이 빈 얇은 원기둥 껍질의 회전 관성

$$\title{Rotational Inertia of Hollow Cylinder}$$ $$r=R,$$ $$ \sigma=\frac{\dd m}{\dd a}=\frac{M}{A}=\frac{M}{LH}=\frac{M}{2\pi R H} \taag1$$ $$ \begin{aligned} l&=r\theta,\\ \dd l&=r\dd\theta\\ &=R\dd\theta\taag2 \end{aligned} $$ $$ \begin{aligned} \dd a&=\dd l\cdot\dd h \\ &=(R\dd\theta)\cdot\dd h \\ &=R\cdot\dd\theta\dd h \\ \end{aligned} $$ $$ \begin{aligned} \dd m&=\sigma\cdot\dd a\\ &=\sigma\cdot (R\dd\theta\dd h)\\ &=\sigma R\cdot\dd\theta\dd h\taag3\\ \end{aligned} $$ $$ \begin{aligned} \dd I&=r^2\cdot \dd m\\ &=R^2\cdot \(\sigma R\dd\theta\dd h\)\\ &=\sigma R^3\cdot\dd\theta\dd h\taag4\\ \end{aligned} $$ $$ \begin{aligned} I_{\text{Hollow Cylinder}}&=\oiint_A \dd I\\ &=\int_0^{H}\int_0^{2\pi} \sigma R^3\cdot\dd\theta\dd h\\ &=\sigma \cdot R^3\cdot \(\int_0^{2\pi} \dd\theta\)\cdot\(\int_0^{H}\dd h\)\\ &=\(\frac{M}{2\pi R H}\)\cdot R^3\cdot (2\pi)\cdot(H)\\ &=MR^2\\ \end{aligned} $$