Intuitive and formal meanings of the definite integral in engineering education

Abstract

The definite integral is a central concept in the applications of calculus to experimental sciences and engineering, and consequently, a relevant didactic research topic. This paper analyses the different meani ngs of the def i nite i ntegral by applying theoretical tools of the ontosemiotic approach to mathematical knowledge and i nst r uction, par ticularly, the interpretation of meaning in terms of systems of operational and discursive practices related to the resolution of types of problems and the algebraization levels model of mathematical activity. Types of problem-sit uations and configurations of practices, objects, and processes are identified that allow us to characterize and articulate the various partial meanings of the definite integral (geometric-intuitive, as a limit of Riemann sums and cumulative function) as well as its extensions to the case of the double integral (as a particular case of the multiple) and the line, from the most intuitive to the most formal. The analysis allows us to identify the generalit y degrees of the objects of integral calculus and the role of algebra in the characterization of the meanings of the definite integral, which must be considered in the planning and management of the processes of teaching and learning of integral calculus in engineering degrees.