Measure of a shape's area relative to a given axis - Can be calculated for any axis using the [[Parallel Axis Theorem|parallel axis theorem]] - Associated with resistance to bending along a reference axis - a long rectangular plank laid on the skinnier side is less resistant to bending than laid wider side down > Second moment of area > $ > I_x=\int_Ay^2\,dA\qquad I_y=\int_Ax^2\,dA > $ > Second (polar) moment of area (by the [[Perpendicular Axis Theorem|perpendicular axis theorem]]) > $ > J=I_x+I_y > $ - $x$ - distance from the y-axis - $y$ - distance from the x-axis ## Shapes ### Rectangle Applying this to a rectangle with base $b$ and height $h$ centered at the origin: $ \begin{align} I_x&= \int_A y^2\,dA\quad &I_y&=\int x^2\,dA\\ &=\int_{-\frac{h}{2}}^\frac{h}{2}by^2\,dA\quad&&=\int_{-\frac{b}{2}}^\frac{b}{2} hy^2\,dA\\ &=b\left[\frac{y^3}{3}\right]_{-\frac{h}{2}}^\frac{h}{2}\quad&&=h\left[\frac{y^3}{3}\right]_{-\frac{b}{2}}^\frac{b}{2} \end{align} $ $ \boxed{I_x =\frac{bh^3}{12}\qquad I_y=\frac{b^3h}{12}} $ ### Circle Given a circle with radius $r$: $ I_x=I_y=\frac{\pi r^4}{4} $