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Topic: functions defined with integrals
Replies: 2   Last Post: Oct 31, 1997 4:31 PM

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 Richard Sisley Posts: 4,189 Registered: 12/6/04
functions defined with integrals
Posted: Oct 30, 1997 2:39 PM

In a recent post Doug Kuhlmann described nicely a class of functions as
follows:

What Joshua seems to be calling indefinite integrals are things like
int(a,x,f(t)dt), i.e. functions defined as definite integrals where the
upper limit is variable. These are indeed very important--I just don't
call them indefinite integrals. I call them functions defined as
integrals. Indeed, I use these very critters, along with numerical
integration techniques, to lead students to discover the FTC.

I would prefer to use Doug's verbal description of such functions rather
than descirbe them as a certain type of integral (an indefinite
integral). I have this preference because I think it will help my
students avoid confusion and keep in the forefront the idea that
integrals are numbers.

I have written some problems illustrating applications of such integral
valued functions to keep track of accumulated changes over lengthing
intervals. But I would like to pose a question to this group. In
Doug's descriptions of these functions "x" would be use as the upper
limit of integration.

If the upper limit of integration is some function of "x" other than the
identity function, the function defined with the integral can be
decomposed for the purpose of writing an expression for its derivative.
With sufficient practice, students can learn to respond with correct
symbol patterns when asked to write an expression for the derivative.
However, I have been unable to think of a credible application in which
such a function naturally occurs. Does anyone have such an example to
offer us? I would be nice to have an example where unit designations
coordinate with the calculations.

Sincerely,

Richard Sisley