# Moment of inertia of figure

Determine the moment of inertia of the area about the x axis.

Answer

$S = int dS = int (from 0 to 1) Y*dx = int(from 0 to 1) sqrt{x}*dx =(2/3)*x^3/2 =2/3 m^2$

Total mass $M = rho*z*S$ $rho$ is density, $z$ is the third coordinate

infinitesimal mass is $dm = M/S *dS = M*(3/2) *Y*dx = (3/2)*M*sqrt{x}*dx$

moment of inertia

$I = int (x from 0 to 1) (Y^2*dM) =int (from 0 to 1) (3/2)*M*(x*sqrt{x}*dx) =$

$=(3/2)*M*(2/5)*x^{(5/2)} (x from 0 to 1)=(3/5)*M$