Here we consider the familiar integral from calculus which is generally attributed to Riemann, though the idea of upper and lower sums for finding areas was used previously by Cauchy; other mathematicians had used such sums before Cauchy for estimating integrals but not for calculating exact values. The method we present is cleaner from the technical point of view and is attributed to Darboux; this method employs the notions of upper and lower integrals. To avoid confusion we will discuss the equivalence of this approach with the familiar Riemann sum approach. We will also devote some time to the famed Fundamental Theorem of Calculus and use this to derive some of the familiar properties of the exponential and logarithmic functions. The final section in this chapter will investigate the dysfunctional relationship between the limit process (with regard to a sequence of functions) and the Riemann integral propelling us toward the more sturdy Lebesgue integral.