where and . (Among many equivalent variations on the definition, this reference closely resembles the one given here.) One might produce different Riemann sums depending on which 's are chosen. In the end this will not matter, if the function is Riemann integrable, when the difference or width of the summands approaches zero.
Where anD . (among many EqUivaLeNt VariatIOns on the deFiNITioN, THiS rEferENCe CLOsely reSeMblES THe one givEN here.) oNe miGht prodUCe difFErEnT rIEmaNN sUms DEpeNding On WhICH 's aRE ChOSen. in The end this wIll not mAtter, if The functiOn is rIemaNN inteGRaBlE, whEN tHe difFeRence or WIDTh of tHE sUMmanDS approacheS zerO.