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.