Integration ( Multi - Variable Included ) from First Principles

Integration has traditionally been so closely linked to the interpretation as an area and to the techniques of anti-differentiation as to appear inseparable from them. While largely a consequence of the fact that, in prePersonal-Computer times, anti-differentiation was the key to effective integration and that line and surface integrals were generally intractable using that technique, the advent of computer algebra systems and easy large scale numerical computation seems not to have had much effect on the way integration is presented in standard texts. On the one hand, it is not clear what sorts of skills are required for a novice to handle computer algebra effectively and on the other hand most available software does not provide the data structures and tools for dealing conveniently with numerical integration from first principles in the general case. We focus on the latter. In this article we examine an approach to the principles of integration based on computer manipulation of multi-dimensional arrays for the coordinate grids, referring to the area interpretation as only one among several possibilities and presenting integration as the solution to non-trivial anti-differentiation problems. Underlying the implementation of the approach is the mathematical notation of Iversons J, an array-processing, functional, computer language. We suggest that the mathematical foundations of the topic – existence of, and convergence to, the limit – should be postponed till after students can effectively compute and manipulate the approximations that are used to define integrals from first principles.