Ways to approach the area under a curve: A study with high school students

In this paper, we characterize how high school students build the idea of approaching the surface area under a curve in a teaching experiment using applets. Tasks were designed taking into account a hypothetical learning trajectory of the definite integral concept, particularly considering the phases of the concept construction supported by reflective abstraction. The analysis of the interactions between students and their answers allow us to identify three characteristics of the construction pro-cess of the idea of approaching the area under a curve: not considering the idea of approaching the area as a limit of a sequence; understanding the approaching to the area by the dynamic and metric conceptions of the limit of upper and lower sums, but not considering the connection between both conceptions; and the construction of the idea of approximation error bound in relation to the number of subintervals of the partition generated by the coordination of the dynamic and metric conceptions of the limit.