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Topic: anti-derivatives, integrals, and units
Replies: 2   Last Post: Nov 10, 1997 2:09 PM

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 Richard Sisley Posts: 4,189 Registered: 12/6/04
anti-derivatives, integrals, and units
Posted: Nov 8, 1997 8:05 AM

I know we had a rather lengthy exchange on anti-derivative versus
indefinite integral terminology recently, but in working on some simple
examples I have discovered something I did not realize before, and I
thought others might find it interesting.

Here is such an example:

Suppose the integrand function is given by 30t and the units are miles
per hour. Problem--find the net change of position over the one hour
time interval [1,2].

By a simple use of the second form of the Fundamental Theorem of
Calculus the answer can be computed by choosing an anti-derivative
expression such as the convenient 15t^2, evaluating it at the endpoints
and subtracting the results in the proper order.

All very routine and basic. However, thing about the units. It seems
to me that the limits of integration should retain their designation as
"hours" as the computation with the anti-derivative is carried out. We
certainly want the computed difference to have units "miles."
Therefore, it seems to me that the anti-derivative expression, 15t^2
should have the same units as the integrand expression (miles per hour)
and therefore a different unit designation than the integral.

Does this not give another reason for reserving the word "integral"
exclusively for the number computed from the Fundamental Theorem (or
approximated in any number of ways)? Since in applied context,
anti-derivatives would retain the unit designation of the rate of change
function, the use of the word "integral" in a phrase to describe an
anti-derivative or a family of anti-derivatives could cause confusion.

Sincerely,

Richard Sisley