Integrais definidas de uma e mais variáveis: uma proposta de intervenção com tarefas exploratórias

Abstract

Considering the difficulties that students in the Differential and Integral Calculus discipline have in understanding the concept of a definite integral of a variable, this research aimed to investigate the elaboration and implementation of an intervention proposal, based on working with problem solving episodes. tasks, which offers CDI students opportunities to explore this concept and, from there, perform generalization movement(s) to understand a multivariational definite integral. We discuss, as a theoretical foundation, the importance of Riemann Integrals and sums with a multiplicative base in the understanding of definite integrals, as well as generalization movements associated with the multivariate integral. We also brought a characterization of the methodology used in our research, as well as the context of intervention and data collection. The analysis of the discussions carried out in small groups about three exploratory tasks is based on a framework that deals with layers of knowledge, with regard to understanding the concept of definite integrals, especially Riemann's Integrals, and restructuring movements/ expansion of the concepts of integrals of one variable. As a result, we could infer that with exploratory tasks 1 and 2, students were able to substantially explore the concept of multiplicative base sum present in the Riemann sum, in relation to the product, sum and limit layers. Regarding the generalization movements provided by task 3, the expansive generalization was used to expand procedural issues of the calculation of an integral and the reconstructive generalization was used in the understanding of structural aspects of the Riemann integral of more variables, such as the use of the geometric structure of the definite integral to structure a way to calculate the mass of an object.