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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:
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.
Lopez Fernandez, Jorge M. and Omar A. Hernandez Rodriguez, "Teaching the Fundamental Theorem of Calculus: A Historical Reflection," Loci (January 2012), DOI: 10.4169/loci003803