Sometime back, I had quite some fun trying to derive the expressions for MOI of different rigid bodies using calculus. However, I had some trouble with the MOI of a solid sphere about an axis through its centre. The problem being that my final answer was markedly different from the original expression ie 2/5MR^2 ,if my memory is correct. My answer ( it contained terms like pi) was somewhat close to the original answer but not close enough (at least for me). So is the original expression just an approximation or do I have to recheck my work ? Also can someone tell me what assumptions are to be made when calculating the MOI of a " thin rod " ?