Let $D$ be a domain, for example, a cylinder (like a glass of water) of height $H=500$, with circular base of radius $R=300$, which contains spheres (cells) with radius $r = 50\times 10^{-6}$ floating in a viscous liquid. $D$ is not full of spheres, but I know that the spheres are uniformly distributed (not in the mathematical sense).
We freeze $D$ and make a slide parallel to the base of $D$ with height $h=5\times 10^{-6}$ (the slide is like a coin with radius $R=300$ and height $h=5\times 10^{-6}$). We can count the number of pieces (slides) of spheres contained inside the "coin" (of course, each slide of sphere inside the "coin" has different size).
I need to estimate how many spheres there are inside of $D$, but I only have one slide of $D$.
How can I estimate how many spheres there are in D? What mathematical tools can I apply? I thought infinitesimal calculus, but maybe some statistical tool is the most appropriate.
If you count $n$ spheres hitting the single slice, there are about $(n/21) \times 10^{8}$ spheres in $D$.
Under the stated assumptions, $D$ is cut into $10^{8}$ slices. Since the density of spheres is uniform, you expect about $n \times 10^{8}$ pieces of sphere among all the slides. A single sphere, however, has a diameter about $20$ times the thickness of a slide, so each sphere hits (roughly) $21$ slides.
(If you can easily distinguish pieces of sphere that pierce the layer versus those that protrude only part way through, you should find about $2/21 \approx 9.5$% partial protrusions among all spherical pieces. That may serve as a useful check on the method.)