Answer to Question #95995 in Calculus for Rachel

Question

A comparison between the Darboux-integral and the Riemann-integral.

Answer 1

The definition of the Darboux integral considers upper and lower (Darboux) integrals, which exist for any bounded real-valued function f on the interval [a,b]. The Darboux integral exists if and only if the upper and lower integrals are equal. The upper and lower integrals are in turn the infimum and supremum, respectively, of upper and lower (Darboux) sums which over- and underestimate, respectively, the "area under the curve." In particular, for a given partition of the interval of integration, the upper and lower sums add together the areas of rectangular slices whose heights are the supremum and infimum, respectively, of f in each subinterval of the partition.

The Riemann integral is based on the Jordan measure, and defined by taking a limit of a Riemann sum