Student Understanding of the Sign of Negative Definite Integrals in Mathematics and Physics

Abstract In a study of student understanding negative definite integrals at two institutions, we administered written survey and follow-up clinical interviews one institution found that “backward integrals”, where the integral was taken from right to left on x -axis, were most difficult for students interpret. We then conducted additional focused backward second institution. Our analysis uses concept image framework recent categorical mathematical sense making. successful using Fundamental Theorem Calculus determine sign an when symbolic expression provided. When considering in graphical context, often had difficulty if they viewed as spatial area, stating area must always be positive. Some able conceptualize $$\Delta x$$ Δ x or $$dx$$ dx difference change thus signed quantity; these view sum smaller pieces, more justifying integral. Students who used amounts similar images making sign. Even progress made either invoked asked explicitly about specific physical context represented by integral, other than area. The provided meaning