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Re: Wolfram, meet Stefan and Boltzmann

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  • Subject: [mg117498] Re: Wolfram, meet Stefan and Boltzmann
  • From: Helen Read <readhpr at>
  • Date: Mon, 21 Mar 2011 06:13:13 -0500 (EST)
  • References: <im4isj$d2o$>

On 3/20/2011 5:54 AM, Andrzej Kozlowski wrote:
> One reason why the words "primitive" and "anti-derivative" are
> preferred to"indefinite integral" is that they emphasise the non-trivial
> nature of the fundamental theorem of calculus. To say that you can
> compute the definite integral by evaluating an indefinite integral at
> the limits of integration and subtracting sounds almost like a
> tautology; to say that you can compute the integral of a function by
> finding and evaluating its primitive, sounds like the profound result
> that it actually is.

That's why I always use the word "anti-derivative" when first teaching 
the concept in Calculus 1. I don't use the term "indefinite integral" at 
all until *after* we get to the Fundamental Theorem of Calculus. Even in 
Calculus 2, I continue to use the term "anti-derivative" much of the 
time, especially when I want to emphasize what we are actually doing.

In Calculus 1, I start off with approximating the area under a curve 
with Riemann sums, and give the class a Mathematica worksheet where they 
practice calculating Riemann sums numerically, and and a "lab" 
assignment where they not only calculate Riemann sums, but also use some 
functions I defined so they can plot the rectangles and get that visual 
connection. Then they do a second lab assignment on what I call the 
"area function" which is defined as the definite integral of f(x)dx from 
(say) 0 to t, that is, the limit of the Riemann sums as \[CapitalDelta]x 
-> 0. I show them how to enter a definite integral in Mathematica, 
emphasizing that it is the limit of the Riemann sums (and not telling 
them how Mathematica or anyone actually calculates that limit). The lab 
has them plot f(x) and area(t), figure out how the graph of f(x) 
predicts where area(t) is increasing/decreasing, concave up/down, etc., 
and notice what happens to area(t) if you change the starting point from 
0 to something else. There's a big text box at the end of the lab where 
they have to summarize all this, and also make note of "any other 
observations they make" about the connection between the area function 
and the original function.

While they are working on the two lab assignments outside of class, we 
finish up the applications of derivatives chapter and introduce 
antiderivatives, always calling them that, and not using indefinite 
integral notation. The students who did not have calculus in high school 
really get it. In that second lab, they will have an aha moment, when 
they realize that area(t) is an anti-derivative for the original 
function, and how cool that is. I try to time it so that they finish the 
second lab a day or so before we arrive in class at the Fundamental 
Theorem of Calculus, and on that day I ask them a lot of questions about 
the second lab, until they they essentially tell me the Fundamental 
Theorem of Calculus.

The students who had calculus in high school tend not to get it -- they 
think "definite integral" is *defined* to be the indefinite integral 
evaluated at the endpoints, and don't understand what I'm even asking 
about. They miss out on the surprise of the Fundamental Theorem of 
Calculus, because they think it is true by definition.

Helen Read
University of Vermont

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