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Even given their nuclear weapons origin, we lowly civil nuclear power engineers still use Monte Carlo methods all day every day. We all make our employers buy big supercomputers or get us access to the national lab leadership class HPC to just pound the hell out of our reactor design problems with random particle transport chains. Sure there are (dramatically) faster deterministic methods that are generally good enough, but Monte Carlo allows you to use exact geometry and not bother too much with the pesky art of computing average nuclear interaction probabilities.
Heck, my buddy at MIT made an open-source Monte Carlo code called OpenMC that's now run by Argonne National Lab [1]. Now everyone can do truly legit reactor design with Monte Carlo!
[1] https://github.com/openmc-dev/openmc
Yeah. I wrote, a long time ago, a small experiment where you can calculate the area of a crazy drawn shape and approximate it's area using monte carlo simulation.
https://github.com/victorqribeiro/monteCarlo
I don't know if I can plug-in here some of the materials I prepared some years ago. It might help to bridge/connect the use of Monte Carlo techniques and its relations with integration:
https://github.com/ChristopheRappold/MCandPython/blob/master...
Also just to mention something else that is very interesting related with MC : "quasi-Monte Carlo techniques" (small example in https://github.com/ChristopheRappold/MCandPython/blob/master... ). For people interested in MC, just take a look to those keywords.