We know that force between two point charges is:
$$\vec{F}=k\ q\ q'\ \dfrac{\hat{r}}{r^2}\tag1$$
From here how shall we derive the equation for force between continuous distributions of two volume charges, i.e.
$$\vec{F}=k\ \int_{q} \int_{q'} \dfrac{\hat{r}}{r^2}\ dq'\ dq \tag2$$
Unlike physicists and physics teachers, I want to derive it in a rigorous way by not using infinitesimals. I know it is pretty simple using the infinitesimal argument. We replace the two point charges $q'$ and $q$ with infinitesimal charges $\ dq'$ and $\ dq$ respectively. Then add up the infinitesimal bits over the volume to obtain $(2)$.
Please give the in between steps to reach equation $(2)$ from equation $(1)$.
Edit: I know the cube is becoming more and more like a point. Still I don't get it. Can someone at least show why $$\lim_{a \to 0} \dfrac{\Delta^2 F}{k\ \Delta q\ \Delta q'\dfrac{1}{r^2}}=1$$ is true (where $a$ is the side of each cube with which we have partitioned both volume charges.).