Filling an area discretely and continuously

ABSTRACT

The literature dealing with student understanding of integration in general and the Fundamental Theorem of Calculus in particular suggests that although students can integrate properly, they understand little about the process that leads to the definite integral. The definite integral is naturally connected to the antiderivative, the area under the curve and the limit of Riemann sums; these three conceptualizations of the definite integral are useful in different contexts and provide students with what it takes to interpret the definite integrals. Research shows that students rarely invoke the multiplicatively-based summation conception of the definite integral although it is essential for evaluating line integrals, surface integrals and volumes. This paper describes a teaching module that promotes understanding as well as activating all three conceptualizations of the definite integral through motivating the accumulation area function and the results in the Fundamental Theorems of Calculus.

KEYWORDS:

• Calculus
• definite integral
• antiderivative
• Fundamental Theorem of Calculus
• area
• Riemann sums
• accumulation area function
• multiplicatively-based summation