Moment of Inertia Tensor Cylinder.

I am computing the [itex]\hat{I}[/itex] - moment of inertia tensor - of a cylinder with height 2h and radius R, about its axis of symmetry at the point of its centre of mass.



I am working in cartesian coordinaes and am not sure where I am going wrong. (I can see the cylindirical coordiates would be the best option here and have computed it correctly in these coordinates, but would like to know where I am going wrong please...)



I have defined the z axis to be in line with the symmetry axis of the cylinder . I am then working in the principal axes s.t non-diagonal elements are 0.



So by symmetry , I can see that Ixx = Iyy.



Computing Izz:



Moment of Inertia tensor formula: [itex]_{vol}[/itex][itex]\int[/itex] dv[itex]\rho[/itex] (r[itex]^{2}[/itex]δ[itex]_{\alpha\beta}[/itex]-r[itex]_{k,\alpha}[/itex]r[itex]_{k,\beta}[/itex])



[itex]\rho[/itex]=M/∏[itex]^{2}[/itex]2h.

x ranges from R to -R, as does y.

z ranges from h to -h.



So Izz= [itex]_{vol}[/itex][itex]\int[/itex] ( x^2 and y^2) [itex]\rho[/itex] dV

where dV = dx dy dx



This yields: 8R^3M/3∏



So a PI is present, so I can clearly see I have gone wrong. I think this might be due to my ranges .



If someone could point me in the right direction, that would be greatly appeacted :).