A perception exists that physics and engineering students experience difficulty in applying mathematics to physics and engineering coursework. While some curricular projects aim to improve calculus instruction for these students, it is important to specify where calculus curriculum and instructional practice could be enhanced by examining the knowledge and understanding that students do or do not access after instruction. This qualitative study is intended to shed light on students' knowledge about the integral and how that knowledge is applied to physics and engineering.

In this study, nine introductory-level physics and engineering students were interviewed about their understanding of the integral.  They were interviewed twice, with one interview focused on and described as problems similar to those encountered in a mathematics class and the other focused on and described as problems similar to those found in a physics class.

These students provided evidence for several "symbolic forms" that may exist in their cognition.  Some of these symbolic forms resembled the typical interpretations of the integral: an area, an addition over several pieces, and an anti-derivative process.  However, unique features of the students' interpretations help explain how this knowledge has been compiled.  Furthermore, the way in which these symbolic forms were employed throughout the interviews shows a context-dependence on the activation of this knowledge.  The symbolic forms related to area and anti-derivatives were more common and productive during the mathematics interview, while less common and less productive during the physics interview.  By contrast, the symbolic form relating to an addition over several pieces was productive for both interview sessions, suggesting its general utility in understanding the integral in various contexts.

This study suggests that mathematics instruction may need to provide physics and engineering students with more opportunities to understand the integral as an addition over several pieces.  Also, it suggests that physics and engineering instruction may need to reiterate the importance, in physics and engineering contexts, of the integral as an addition over several pieces in order to assist students in applying their knowledge about the integral.