Here is an applet for trying a Monte Carlo simulation to determine the area under a quadratic curve. http://www.dartmouth.edu/~chance/teaching_aids/probability_book/bookapplets/chapter2/MonteCarlo/MonteCarlo.html

How would one compute an estimate of
by such a method?

Method 1: Take points in the unit square, what is the probability
that they fall in the inscribed circle?