Evaluating Integrals Using Self-Similarity

There are more advanced methods, especially Cauchy's residue calculus using complex variable theory, but a typical calculus course mentions only these two methods. Students quickly learn from experience that Archimedes really was a genius, but for practical purposes the Fundamental Theorem is the way to go. But there is a third method, which is quite elementary although not well known. It is derived from the theory of integration on fractals, and is based on a self-similarity property of the unit interval. It is not a truly practical method, since it gives exact answers only for integrals of polynomials, but it illustrates the important mathematical article of faith that the study of exotic structures can produce new insights into old and commonplace subjects. After presenting the method in the context of ordinary integrals, I indicate how it can be adapted to the context of integrals on fractals, where it is essentially the only method available. It is necessary to assume that the integral has a few elementary properties: (i) linearity,