From: Victor Miller <victorsmiller@gmail.com> I would expect that the area of the region is given by an elliptic integral. Use Green's theorem to show that the area is integral y dx where the integral is over the path of the bounding curve. The bounding curve is binational to an elliptic curve, and so is parametrized by elliptic functions.

--uh... really? I thought that general cubic curves are NOT birational to elliptic curves. But no: http://en.wikipedia.org/wiki/Algebraic_curve#Elliptic_curves says they are. Oh. That suggests to get an area that is not expressible in closed form (if elliptic functions are allowed in the closed form) we need an algebraic curve of degree 4. These are in general NOT birational to elliptic curves since they generally have genus 3, while elliptics have genus 1. It is true that y^2 = quartic(x) if has a rational point is birationally elliptic, but these are not general curves. Anyhow, back in my original conjecture, I presume no closed form is possible if we do not allow elliptics in the formulas. I further presume that general elliptic functions evaluated at rational arguments in general yield numbers not expressible in lower-tech ways,