It is customary in an account of Riemann integration to give one or two examples of the evaluation of definite integrals directly from the definition of the integral in terms of Riemann (or Darboux) sums, but the method is used only to illustrate the definition, and is never fully exploited. The method is, however, capable of a more systematic treatment than one might at first expect, and in this note we show how it can be applied to some of the elementary functions of analysis.