Starting from the sum: I=Ʃ mα*ρα2 and replacing it by the appropriate integral, find the moment of inertia of a uniform thin square with side length 2*b, lying in the x-y plane, rotated about the x-axis. Calculate its moment of inertia.
The integral form turns to I=∫ρ(r)*r2dV
The Attempt at a Solution
I rotated it around the z-axis, giving me a cylinder with radius 2b and height 2b. I know the volume of this cylinder is 2*∏*r2*h. And that ρ=mass/volume. I need help in setting up and evaluating the integral because what I got seems wrong.
I used cylindrical coordinates, claiming that:
θ goes from 0 to 2∏
r goes from 0 to 2b
z goes from 0 to 2b
and my volume element dV=r*dr*dz*dθ. I end up with mass*b2, and I do not believe this is correct. Help please?