A Simple Theorem on Riemann Integration, Based on Classroom Experience

At the Georgia Institute of Technology, a computer program is used in freshman calculus which graphically illustrates upper and lower Riemann sums and generates values of their differences. The students often observe that the differences Δn seem to be proportional to 1/n, where n is the number of subdivisions; but this is only approximate. We make this rigorous by showing that Δn = V/n + 0 (1/n3) as n → ∞, for nice functions, where V is the total variation. The proof is simple, and is a nice illustration of the ideas of asymptotic analysis, and several other techniques of analysis.