Advanced Calculus: MATH 410 Riemann Integrals and Integrability

provided it was defined, was a number equal to the area under the graph of f over [a, b]. You also likely learned that the definite integral was defined as a limit of Riemann sums. The Riemann sums you most likely used involved partitioning [a, b] into n uniform subintervals of length (b− a)/n and evaluating f at either the right-hand endpoint, the left-hand endpoint, or the midpoint of each subinterval. At the time your understanding of the notion of limit was likely more intuitive than rigorous. In this section we present the Riemann Integral, a rigorous development of the definite integral built upon the rigorous understanding of limit that you have studied earlier in this course.