ABSTRACT

This chapter discusses the integral concept and builds approximating sums by partitioning the underlying interval and then evaluating the given function at selected sample points in each subinterval. Such sums will be called Riemann sums. When examining the Darboux integral, it was often worthwhile to refine a partition, due in a large part to how this refinement would affect the upper and lower sums. Because Riemann sums have an associated choice of tagging, one does not get as interesting a comparison when looking solely at refinements. Instead, the chapter examines the impact of reducing a partition’s mesh size. One of the main objectives of the chapter was to show that the Riemann integral is the same as the Darboux integral.