The definition of integral

We will now formalize the process outlined in the preceding section in order to define the integral of a function. Until we say otherwise, D is a subset of the real numbers, f is a bounded, real valued function with domain D, and $[a,b]\subset D$. The goal is to define the integral of f on the interval [a,b]. The route will involve sums of the sort encountered in the Preliminary Fundamental Theorem of Calculus.