A non uniform rod AB has a mass M and length 2L. The mass per unit length of the rod is mx at a point of the rod distant x from A. Find moment of inertia of this rod about an axis perpendicular to the rod(a) through A (b) through the mid point of AB.

let the rod is placed along x axis and  its end A is at origin........

consider a small element of thickness dx at a distance x from end A..

mass of element is (mass per unit length)(length of this element)

              dm=mxdx........1

moment of inertia of this element about the axis passing through its  end A and perpendicular to rod is dmx^2

 dI=dmx^2

     =mx^3dx

total moment of inertia is integral of dI limit from 0 to 2L

I=integral(mx^3dx)    lim 0 to 2L

 =m (2L)^4 /4

I =4mL^2