Is Calculating Moment of Inertia for a Disc with (x+dx)^2 Valid?

[Arup Biswas]

In the picture the question marked spot is confusing me. Should the MI be calculated in this ugly way? That we do not consider the points in between and on dx! Why only x is taken as the distance? You may say take x+dx, resulting dx in the exapnsion and the dx from the mass to square and being negligible. But that too is not so simple. Can we write the whole mass as there is and just multiply it by (x+dx)^2. The definition does not say so. Please explain.

 

[Doc Al]

It's not quite clear what you are asking.

Can we write the whole mass as there is

The whole mass of what?

and just multiply it by (x+dx)^2

Ah, I think you're saying: Can you take the mass of the thin ring and multiply it by its radius squared to get its moment of inertia. Sure. In fact, that's just what they are doing. But you want to do it in a way that you can calculate the moment of inertia of the entire disk or a non-infinitesimal ring.