Proving $\tau = I\alpha$ for Continous Mass Distribution

[pardesi]

how does one prove $$\tau=I\alpha$$ for continious mass distribution where $$\tau$$ is the net external torque along the axis of rotation $$I$$ is the moment of inertia,and $$\alpha$$ is the angular accelaration ...
i know the proof when the mass distribution is discrete...

 

[mjsd]

look at your discrete version and see how you can turn the sum into an integral ... eventually, i think the integral get absorbed in the definition of I (moment of inertia)

 

[pardesi]

well taht doesn't happen ...because the discrete version necissates the existence of point like particles...what does ahppen that thsi turns out to be a very good approximation...using the fact that as the mesh value of the riemann sum decrease it converges to the riemann integral; for a closed bounded function