The area inside an algebraic curve of degree<=2 can be computed using well known functions. QUESTION: For general degree=3 curves, is it impossible to express area in closed form? For general-degree general curves, no closed form expression is possible in general... one cheesy proof being: consider P(x)*Q(y)>=0 where P,Q, are polynomials, and P is an unsolvable quintic, while Q(y)=y*(1-y). Then the area of the component is the difference between two successive of the quintic's roots, and we can make the quintic have exactly 3 real roots so this is only finite component. The reason I call that proof "cheesy" is that the impossibility is really a lot more serious than just the inability to solve quintics -- very highly transcendental functions woud be needed to express area in general I think, e.g. hyperelliptic integrals, etc. Anyhow, somebody more versed than I in the theory of what integrals can be done in closed form and what cannot, could possibly answer my degree-3 question above.