The summation that underlies the integral to find a volume, as you noted, assigns a thickness. Area times arbitrarily small thickness dx is an arbitrarily small volume. So, you’re adding an infinite number of infinitely small volumes to get a finite volume. That works. The example I was responding to was adding an infinite number of masses that aren’t arbitrarily small. Hence the problem.

I’m well aware that a 4D density is not the same as a 3D density. That is, in fact, the whole point of the post I made. I order for the post I was responding to to make any sense, the each 3D mass would need to be multiplied by a one dimensional density and an arbitrarily small length. This would have made each summand an arbitrarily small 4D hypermass, and the result would have been a finite 4D hypermass.

Basically, an integral needs a dx, a dr, or a dsomething, and the post I was responding to had none.