가는 막대의 회전 관성

$$\title{Rotational Inertia of Rod}$$ $$ \lambda=\frac{\dd m}{\dd l}=\frac{M}{L}\taag1$$ $$ \dd m=\lambda\cdot\dd l $$ $$ \begin{aligned} \dd I&={r}^2\cdot \dd m\\ &={l}^2\cdot (\lambda\cdot\dd l)\\ &=\lambda{l}^2\cdot \dd l\taag2\\ \end{aligned} $$ $$ \begin{aligned} I_{\text{Rod}}&=\int_L \dd I\\ &=\int_{-\frac{L}{2}}^{\frac{L}{2}} \lambda{l}^2 \dd l\\ &=2\cdot\lambda\cdot\int_{0}^{\frac{L}{2}} {l}^2 \dd l\\ &=2\cdot\(\frac{M}{L}\)\cdot\(\frac{L^3}{24}\)\\ &=\frac{1}{12}ML^2\\ \end{aligned} $$