Investigating Student Interpretations of Integration: dx as punctuation vs. dx as an infinitesimal

Holding a certain conceptualization of an integral may change how someone frames and solves an integral problem. In math courses, notational conventions present the differential dx as a sort of punctuation for an integral. This contrasts with views that dx as an infinitesimally small bit of x, which is more advantageous in physics course contexts. We provide two examples of non-conventional integral questions from semi-structured think-aloud interviews of second-semester Intro-Physics students. In a prior interview question, students did not object to a lack of parentheses surrounding terms in an integrand, despite conventions used in Calculus textbooks. For an integral with a differential in the middle of an expression, differing student utterances and answers led us to define two conceptual resources concerning infinitesimals. We connect these two resources as corresponding to the limit or infinitesimal definitions of integration. This work highlights subtle differences in students' interpretation of integration and future areas for study.