How would a mathematician solve the following physics problem without using the infinitesimal approach?

I was solving a physics problem and I wondered how could it be done using a more formal approach and without resorting to the mindset of infinitesimals

When it comes to calculating the centre of mass of a continuous object, let’s say a cylinder of radius $ r $ and length $ L $ with variable density $ rho(x) $ at each point of the cylinder. (and taking $ x = 0 $ as the left side of the cylinder). I go through the following thought process.

Starting from the formula for the center of mass of a continuous body of mass:

$$
x_{cm} = frac{int x , dm}{int , dm}
$$

I think of dividing the cylinder into very small layers, and one layer has mass $ dm = rho(x) dV(x) $

In this case, the volume of that small layer of the cylinder is $ dV = pi r^2 dx $ where $ dx $ is the very small width of the layer. So $ dm = rho(x) pi r^2 dx $

I replace $ dm $ in the original formula:

$$
x_{cm} = frac{int_0^L x rho(x) pi r^2, dx}{int_0^L , rho(x) pi r^2 dx} \
x_{cm} = frac{int_0^L x rho(x) , dx}{int_0^L , rho(x) dx}
$$

And I keep going about solving the problem doing the integration.

In all this case, I have used Calculus but I have gone against something that I have read from many Calculus books, saying that $ dx $ in the integral is a meaningless operator to determine the variable of integration. However, in this case, thinking of $ dx $ as a very small width helps a lot solving the problem.

I’m aware of the historical context of Calculus and also how engineering and physics majors use this constantly but from a mathematical point of view is not formal.

I’m curious about how a mathematician would solve this problem if he doesn’t want to use the "physicist" approach of treating $ dx $ as an infinitesimal and something "active" in the calculations?