you could use curvePoint() (or bezierPoint(), depending on type of curve) to decimate the curve into a polygonal approximation, then calc polygon area (google or wiki for it, it's simple: abs(sum(v[i] X v[i-1])/2) in pseudocode), increase decimation as necessary to better approximate

You can also count the pixels of the color of the curve, supposing it is filled.

Short answer: yes.

Depending on the kind of curve, you can either do basic analytical integration using typical Java math operations or, as the previous posters have described, use Processing's graphics functions to derive useful approximations.

kb, http://www.riftgame.com/

Please post some examples!

Throws a curveball - how about the area bounded by a brownian island  http://www.cs.princeton.edu/introcs/23recursion/BrownianIsland.java