Areas, anti-derivatives, and adding up pieces: Definite integrals in pure mathematics and applied science contexts

•Three conceptualizations of the definite integral are analyzed in mathematics and science contexts.•All three conceptualizations are productive for decontextualized mathematics integrals.•Those based on area under a curve and anti-derivatives are less productive in applied contexts.•The Riemann sum-based adding up pieces conceptualization is highly productive in applied contexts.

Research in mathematics and science education reveals a disconnect for students as they attempt to apply their mathematical knowledge to science and engineering. With this conclusion in mind, this paper investigates a particular calculus topic that is used frequently in science and engineering: the definite integral. The results of this study demonstrate that certain conceptualizations of the definite integral, including the area under a curve and the values of an anti-derivative, are limited in their ability to help students make sense of contextualized integrals. In contrast, the Riemann sum-based “adding up pieces” conception of the definite integral (renamed in this paper as the “multiplicatively-based summation” conception) is helpful and useful in making sense of a variety of applied integral expressions and equations. Implications for curriculum and instruction are discussed.