Is it possible to calculate volume of a cube (with volume $L^3$) by filling it with small balls each with a radius $r_N$ and the balls are disjoint.
Let the number of balls be $N$
$$\lim\limits_{N\mapsto \infty} r_N=0$$
Is it always possibles to have $$\lim\limits_{N\mapsto \infty} N \frac {4\pi r_N^3}{3}=L^3$$ $\quad$
Please keep in mind we are not talking about integration in spherical coordinates .
I know Radon-Nikodym theorem see Radon-Nikodym theorem allows us to calculate the volume of the cube using small balls but not necessarily disjoint and it is not obvious that the radius of each ball is the same .
Your problem is the same as filling balls of unit radius into cubes of larger and larger side length and taking the limit as the side length goes to $+\infty$. This converts your problem into the Kepler conjecture.
The theorem of Hales and co-authors, saying that the Kepler conjecture is true, shows that the ratio between the left and right sides of your conjectured equation is no greater than $$\frac{\pi}{3 \sqrt{2}} = 0.740480489... $$