DEFINITE INTEGRALS, RIEMANN SUMS, AND AREA UNDER A CURVE: WHAT IS NECESSARY AND SUFFICIENT?

A teaching experiment was conducted in a calculus class to determine what it means to understand definite integrals. One interesting result was based on students ' use of area under a curve as a tool for computing definite integrals. Results show that in the problems presented in this study, students ’ use of area under a curve was helpful in problem solving only when a deeper understanding of the structure behind the definite integral was present. The purpose of this research was to examine student understanding of Riemann sums and definite integrals. These concepts are imperative for students to understand for three main reasons. First, many real world applications involve functions that do not have an antiderivative that can be expressed in terms of elementary functions. For example, the antiderivative of the function 2) ( xexf = cannot be expressed in terms of elementary functions. Thus, the Fundamental Theorem of Calculus could not be applied, and other methods for evaluating the definite integral, such as Riemann sums would be needed. This leads to the second reason that students need to have an understanding of the structure of Riemann sums. While Riemann sums may not be the most efficient method fo