How to derive magnetic moment for integrated circuit level?

Imagine we have a circuit on the xy-plane, with a random geometry (it just need to be closed). I want to calculate the magnetic moment of this setup:

$$vec{m} = frac{1}{2} int_V d^3 x’ vec{x}’ times vec{j}(vec{x}’)$$

Assuming that current is constant we get $I d vec{x}’ = vec{j}(vec{x}’)$ which turns the integral into:

$$vec{m} = frac{1}{2} int_V d^3 x’ vec{x}’ times (I dvec{x}’)$$

Next step in the computation, I’m trying to follow, is immediately writing down:
$$vec{m} = frac{1}{2} I int_{delta A} vec{x}’ times dvec{x}’$$

I probably flunked an important lesson and can’t understand what is happening here. We don’t use Stokes, because then the “$times$” would go away.

My question is: how does the integral over the volume/surface transform into one over its edge?