This technique should be straightforwardly adaptable to compute arbitrary moments, not just the centroid.

If you have a scalar-valued function that you can conveniently express as the divergence of any closed-form function, you can integrate it like this. And you can generalize beyond scalar-valued functions and beyond Euclidean space using the generalized Stokes’ theorem.

You can even do this in real life: if you want to integrate the electric current density through a surface (that is, measure the total current crossing the surface), you can integrate its anti-curl (is that a word?) around the boundary of that surface, which is what a current transformer or a clamp-on current meter does.

I bet there’s a hydraulic or pneumatic analog as well, but a nontrivial example isn’t immediately coming to mind.