Math Idea – Stochastic Integration?

A quick math idea.

I was wondering. If you take just a limited amount of samples from the function. Could you calculate the integral? Like a stochastic version of integration.

The idea is that this could bypass ?any problematic part. For instance in integrating exp ( -x² ).

So as I use a bunch of random but limited amount of samples as input. I assume I also get a Gaussian distribution as output but with a narrow band.

That is. I guess my primitiv function will have some sort of distribution. A Gaussian that produces a Gaussian maybe.