Students’ conceptions of double integral bounds: an APOS theory perspective

In this article we use the Action, Process, Object, Schema framework as a theoretical lens to analyse students’ responses to an examination question in a multivariable calculus course for engineering majors. The exercise required students to translate between a geometric representation of a lamina into two different symbolic representations of the area of the lamina as double integrals in the Cartesian coordinate system. We designed and empirically refined a genetic decomposition of double integral for area and used this as an analysis tool to capture and characterise students’ conceptual understanding. A notable finding is that when the region needs to be partitioned to correctly describe it using double integrals, the success rate is significantly lower than when the region does not need to be partitioned. We present five different forms of incorrect double integral constructions and identify two primary elements of conceptual understanding which are necessary for successful construction of an iterated double integral for the area of a lamina.