Question

Calcaluting mass with density different integral methods different results?

Answer 1

Use one for spherical objects, the other for cylindrical objects. I believe that is the simple answer you are looking for.

Answer 2

You have the integralM=∫2ππdθ∫21r3sin2θdrthis seems fine. One thing is sure this integral is not zero: Indeed your can write it as a product of two integralsM=(∫2ππsin2θdθ)(∫21r3dr)

and both those integrals give strictly positive numbers.

I would advise you to compute these two integrals separately and check you get stricly positive numbers.