Teaching the Fundamental Theorem of Calculus: A Historical Reflection - Teaching Advantages of the Axiomatic Approach to the Elementary Integral

The axiomatic approach to the integral provides some clear teaching advantages when compared to the usual presentation using the Cauchy limit-sum definition of the integral. The latter presentation requires an understanding of a complicated type of limit. Whether the limit is to be understood in the sense of convergence of nets with respect to the directed order defined by refinement (inverse set inclusion), or as a limit when the “norm” of the partitions goes to zero (see note 7.1), the limit, given its complicated nature, presents obvious and sometimes insurmountable difficulties for students.  We list some of the advantages to teaching the elementary integral using Gillman's axiomatic approach:

• The properties of the integral are easily justified and readily obtained. Furthermore, both versions of the FTC are central ingredients of the presentation.
• The use of L. Gillman’s [20] heuristics and Gillman's limit bounding condition (B’) makes the setting up of integrals (usually justified through Riemann sums) an easy and attainable matter. The condition of additivity is obvious for all applications of the integral.
• The theoretical importance of Darboux integration becomes clearer in the axiomatic approach. In this approach one supposes that there is an integral for each continuous function on a given interval of the real line. In the second half of the seventeenth century, mathematicians did not contemplate the need for a justification of such a statement. But as mathematics progressed, this supposition eventually required a theoretical justification. Eventually, the Darboux approach to integration as well as Cauchy's limit-sum property led to the modern theories of integration we know today.
• If we are to believe that the history of mathematics somehow reflects “cognitively correct ways” to approach the teaching of mathematics, then the axiomatic presentation of the elementary integral is in total harmony with the historic evolution of the notion of integration.

Note for page 7:

7.1. Remember that the choice of points at which the function is evaluated must be taken into account in taking the limit. The important result that such a limit is independent of the choice of points is an idea that needs time to mature and assimilate.