A good def for the R. integral is: the one number which is both a lower bound of the set of all upper sums (area with superscribed rectangles), and an upper bou

A good def for the R. integral is: the one number which is both a lower bound of the set of all upper sums (area with superscribed rectangles), and an upper bound of all lower sums (inscribed rects). That has a lot of nice properties which are useful in proofs, whereas the limit of sums definition suffers from there being too many ways for "delta x goes to 0" which need to be shown to be equivalent at some point to be useful (nasty!).