In reply to Gerwin Hassink.

Thanks for writing. You ask good questions.
1. In fact, the random number generator (RNG) I am using returns uniform variates in the open interval (0,1), so I should have written (0,1) instead of [0,1] in several places.
2. For area estimation, it doesn't matter whether you integrate the quarter circle function on the closed integral [0,1] or the open interval (0,1). The area under the unit circle is pi/4 whether you include the axes (and origin) or not. This is because the axes (and origin) are sets of "measure zero" in the 2-D plane.
3. From a Monte Carlo perspective, one or two extra points "inside" the circle are not going to change the estimate much compared to the standard error of the estimate. So it doesn't matter how you count a point that is on an axis. For that matter, you'll get a similar estimate whether you count a point as "in the circle" when x^2 + y^2 < 1 or whether you count "in" as x^2 + y^2 ≤ 1.