Forgive me if I'm being naive, but, I don't understand why the $x$-coordinate of the Centre of mass is taken as an integral of $x.dm$ and not $m.dx$?
I understand the summation part, but how do we convert that into an integral? It could very well be a mathematics question, as the fundamentals of calculus, since it belongs in the domain of physic, I thought might as well.
Besides isn't the $x$-coordinate the independent quantity just as time is the independent quantity for when we write acceleration as the integral of $v.dt$? This is really troubling!