Question 426974
Let {{{S[L] }}}= sum of the areas of the inscribed rectangles (also called the lower Riemann sum), and 

{{{S[U]}}} = sum of the areas of the circumscribed rectangles (also called the upper Riemann sum).

Then if the curve {{{y = f(x) >= 0}}} over the interval [a,b], (and using the same partitioning of the interval) then {{{S[U] >= S[L]}}}.  As the partitioning gets finer (i.e., more subintervals), the two sums should converge to a common value which is the definite integral of f(x) over [a,b] if y = f(x) is integrable over [a,b]. (Note that if {{{y = f(x) <= 0}}} then the relationship between the two Riemann sums is also reversed.)