속이 채워진 직사각형의 회전관성

$$\title{Rotational Inertia of Rectangle}$$ $$ r^2=x^2+ y^2,$$ $$ \sigma=\frac{\dd m}{\dd a}=\frac{M}{A}=\frac{M}{X Y}\taag1$$ $$ \dd a=\dd x \dd y\taag2 $$ $$ \begin{aligned} \dd m&=\sigma\cdot\dd a\\ &=\sigma\cdot(\dd x \dd y)\\ &=\sigma\cdot\dd x \dd y\taag3\\ \end{aligned} $$ $$ \begin{aligned} \dd I&={r}^2\cdot \dd m\\ &=(x^2+ y^2)\cdot(\sigma\cdot\dd x \dd y)\\ &=\sigma(x^2+ y^2)\cdot\dd x \dd y\\ \end{aligned} $$ $$\put L^2=X^2+ Y^2,$$ $$ \begin{aligned} I_{\text{Rectangle}}&=\iint_A \dd I\\ &=\int_{-\frac{Y}{2}}^{\frac{Y}{2}}\int_{-\frac{X}{2}}^{\frac{X}{2}} \sigma(x^2+ y^2)\cdot\dd x \dd y\\ &=4\sigma\int_{0}^{\frac{Y}{2}}\int_{0}^{\frac{X}{2}} (x^2+ y^2)\cdot\dd x \dd y\\ &=4\cdot\sigma\cdot\int_{0}^{\frac{Y}{2}}\(\frac{X^3}{24}+\frac{X y^2}{2}\) \dd y\\ &=4\cdot\(\frac{M}{XY}\)\cdot\(\frac{X^3 Y}{48}+\frac{X Y^3}{48}\)\\ &=\frac{1}{12}M(X^2+Y^2)\\ &=\frac{1}{12}ML^2\\ \end{aligned} $$