Is the validity of measuring area by approximation an assumption of calculus? The assumption that if you subdivide an area into more and more sub intervals, the approximation gets better and better. Has this been formally proved, or is it just intuition? Thanks!
 A: I think a good thing for you to think about is upper and lower Riemann sums. 
If you have an over-approximation of the area (via the assumption of finite additivity of area) and an under-approximation of the area (again via this assumption), and the over-approximation and under-approximation converge to the same thing then we can safely call/define this the area.
These upper and lower sums don't always converge in which case we have to rethink things.
A: Anon beat me to the point in the comments above :)
In measure theory, there is a notion of regular measures of which the Lebesgue measure is an example. Being regular essentially says that (measurable) sets can be approximated in this way.
Now if you're thinking of things like integrals, remember that integrals are based upon measures, and that a function is integrable only if it satisfies some restrictions. Only certain functions are Riemann integrable, for example, precisely because they are susceptible to approximation. 
Lebesgue integrability is the same way: a function has to be Lebesgue measurable and Lebesgue integrable in order to form the integral, and that means by definition the signed area underneath can be approximated, and hence the area is defined to be that approximation.
On the other hand, I think it's likely (although I'm no historian) that the ancients may have had some Platonic notion of area of plane figures, and that they took for granted that this area was equal to approximations. This idea could be wrong, however. For one thing, they managed to navigate irrational numbers in ruler-compass constructions without the real numbers. They did so using only the notion of length as "magnitudes" which were not necessarily rational multiples of their units of measure. Perhaps they did something simliar with area...
