## Abstract

Students use a variety of resources to make sense of integration, and interpreting the definite integral as a sum of infinitesimal products (rooted in the concept of a Riemann sum) is particularly useful in many physical contexts. This study of beginning and upper-level undergraduate physics students examines some obstacles students encounter when trying to make sense of integration, as well as some discomfort and skepticism some students maintain even after constructing useful conceptions of the integral. In particular, many students attempt to explain what integration does by trying to use algebraic sense-making to interpret the symbolic manipulations involved in using the Fundamental Theorem of Calculus. Consequently, students demonstrate a reluctance to use their understanding of “what a Riemann sum does” to interpret “what an integral does.” This research suggests an absence of instructional attention to subtle differences between sense-making in algebra and sense-making in calculus, perhaps inhibiting efforts to promote Riemann sum interpretations of the integral during calculus instruction.

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## Notes

There are, of course, entirely valid ways of extracting conceptual sense of the definite integral by interpreting antiderivatives of the integrand. These do not, however, make use of Riemann sum reasoning, nor do they shed light on the symbolic manipulations involved in finding the antiderivatives.

One beginning physics student, B1, showed sporadic use of Riemann sum-based reasoning, but did not use it consistently and he could not decide on the units for The Integral of Position Problem (see below). Primarily for reasons of space, this case will not be examined here, but I will use the language that “all beginning students” failed to use this reasoning strategy for ease of presentation.

A few beginning students occasionally made mention of sums or rectangles while discussing the definite integral, however none of them could successfully

*use*them to explain what the integral did or what its units were.The area chosen by U5 lends itself to a more general geometric analysis that reveals algebraic sense behind the antiderivative area calculation as the difference of areas of triangles. This geometric argument does not hold for polynomials of higher degree, however, nor does it shed any light on why the algebraic representation should correspond to the antiderivative of the bounding function.

Both of his interpretations, with some careful tweaking, were quite reasonable, but the details are outside the scope of this paper.

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## Acknowledgements

This research was supported in part by the National Science Foundation under grant No. PHY-1405616, and in part by a faculty development leave from Xavier University. Any opinions, findings, conclusions, or recommendations expressed in this paper are those of the author and do not necessarily reflect the views of the National Science Foundation or Xavier University. I would like to thank Corinne Manogue, Tevian Dray, John Thompson, and Mike Loverude for their thoughtful conversations and feedback throughout the course this research.

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Wagner, J.F. Students’ Obstacles to Using Riemann Sum Interpretations of the Definite Integral.
*Int. J. Res. Undergrad. Math. Ed.* **4**, 327–356 (2018). https://doi.org/10.1007/s40753-017-0060-7

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DOI: https://doi.org/10.1007/s40753-017-0060-7