What is the advantage of this Monte Carlo approach over a typical numerical integration method (like Runge-Kutta)?

As a hobbyist, I'm playing with analog computer circuits right now. If you can match your curve with a similar voltage profile, a simple analog integrator (an op-amp with a capacitor connected in feedback) will also give you the area under the curve (also as a voltage of course).

Analog circuits (and op-amps just generally) are surprising cool. I know, kind of off on a tangent here but I have integration on the brain lately. You say "4 lines of Python", ad I say "1 op-amp".)