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Reasoning with "dx" in the integral

Robert Ely*, University of Idaho
Talk Abstract: 
Research indicates that in order for students to be able to successfully interpret and flexibly model with definite integrals, they must conceptualize the integral as “adding up the pieces” of a quantity, rather than as (a) a symbolic template, (b) an anti-derivative, or even just (c) an area under a curve. Task-based interviews with students in my recent experimental calculus class indicate that the “adding up the pieces” conceptualization of $\int_a^b f(x) dx$ requires the student to be able to view the “f(x)dx” as a small bit of the total quantity. This understanding in turn relies on being able to treat “dx” as an increment of x, which has a size, across which the quantity to be summed accumulates (at a rate of f(x)). I argue that this incremental understanding of dx was also crucial in allowing Leibniz to develop calculus where his predecessors (e.g. Cavalieri and Torricelli), who used indivisibles that have no “width,” did not.
Time Slot: 
Saturday, April 2, 2016 - 10:35
Room Number: 
STAG 262