[math-fun] area inside algebraic curve impossible to express?

For an explicit impossibility conjecture, consider the finite-area loop containing (-2,1) bounded by the fairly-generic-looking cubic curve 3x^3 + 4x^2y + 7y^2x - 10y^3 - 36x + 37y = 41. I doubt this area, which is approximately 4.68450311946587, can be expressed in closed form. [Number found using polar coordinates centered at (-2.2, 1.1); integrate (r^2/2) dtheta using trapezoidal rule. All digits probably correct.] If anybody recomputed this to, say, 50 decimals, then huge runs of number-guesser tools could partially confirm the conjecture, I guess. Ries -l6 realizes it has nothing & gives up in disgust. -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)