The Riemann Integral

Abstract

It is common, at an elementary level, to define integration as the process that is the inverse of differentiation. Thus, given a function f, we look for a function F such that F′(x) = f(x) for all x. There is, in fact, no reason why such a function should exist. Next, if a and b are real numbers with a < b, it is shown that F(b)−F(a) gives the value of what is intuitively under-stood to be the area under the curve y = f(x) between x = a and x = b. This approach is unsatisfactory first because it gives no meaning to ‘integral’ for a function which is not a derivative, and second because it cannot cope with situations where the notion of area under the curve has no obvious intuitive meaning.