TITLE: Have people tried to find the area under a curve by means other than integration?
QUESTION [5 upvotes]: It seems pretty frustrating that some definite integrals can only be evaluated numerically, so have people tried to find another method of finding the area under a curve that isn't numerical? I'm talking about a method that gives a function that can be evaluated in order to find the area. Is there only one such function for a given curve (the function given by integrating)? Thanks.

REPLY [2 votes]: Monte Carlo integration methods can be adapted to be "non-numerical." The idea boils down to randomly sampling a curve bounded by a much simpler one (e.g., a rectangle) at many discrete points. One then counts how many samples were inside the curve and how many were outside. Knowing the area of the simpler bounding shape, the area of (or under) the curve can be estimated. 
The classic example of this is throwing darts (but it could be dropping pins or marbles) at a circle inscribed within a square. Often simple scheme is used to estimate $\pi$ (very inefficiently), but it of course also estimates the area of the circle:

Monte Carlo techniques, in their numerical forms, are important for things such as rejection sampling to obtain random numbers from a desired distribution.