<< 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. >>
Here is a quick attempt. Is this what you had in mind? Okay its not really "practical" but ...
The area of a square is given by A(s) = s^2 where s is the length of the side. s in feet; A(s) in sq. ft. Then dA/ds = 2s is in sq.ft / ft.
Suppose that s is a function of t (time) say s(t) = sin(t) (in ft.) (Perhaps t in hours is the time of day and the side is being affected by the temperature which goes up and down as the day goes on.)
First: A(s(t))= (sin(t) )^2 Units = sq. ft.
But also A(t) = Integral( 2x dx, 0, s(t)) With x being, of course, a "dummy" variable. That is A(t) is an accumulation function which gives the accumulated (net change in) area as the the sides varies from zero to s(t) (its length at any time t). Then
A(t) = Integral( 2x dx, 0, s(t)) = x^2 | 0 to s(t) = (s(t))^2 - s(0)^2 = (sin(t))^2 as above. Units: The integrand 2x dx is is sq.ft / ft. The antiderivative x^2 in sq.ft/ft times ft = sq ft.
Is this what you had in mind? Comments? Questions? Corrections? Improvements?