7.2.1 Speedometer and Odometer

In Section 7.1 we worked very hard to solve the problem of finding distance from velocity and then to abstract the procedure into the concept of definite integral, which we can use to find average values of continuous functions. But we already knew how to get a distance function from a velocity function by a somewhat simpler process, that of antidifferentiation. What is the connection between the two ideas, integration and antidifferentiation?

We can answer this question by looking again at the distance-from-velocity problem, for which our two answers appear to be very different. The fact that they are the same will show us what the connection must be — not just in that specific context, but in a setting that applies to any continuous function.

Recall that, if `stext[(]t text[)]` is the function giving the distance traveled at any time `t` after a starting time `t=a`, then `stext[(]t text[)]` is the solution of the initial value problem

The total distance traveled from time `t=a` to time `t=b` is just `stext[(]btext[)]`. If we know the function that solves this initial value problem — a function that is a particular antiderivative of `v` —then we know the distance traveled at any time between `a` and `b`.

The odometer (center)
reads 89,582 (and a
fraction) miles, and the
trip meter (top) reads
505 (and a fraction) miles.

We can set the trip meter in our car to zero at the start of the trip, and the meter will record `stext[(]t text[)]` as we travel. In particular, its reading at the end of the trip will be `stext[(]btext[)]`. On the other hand, we don't actually need to know the distance function to find distance traveled. That is, instead of using a trip meter, we can use the odometer. Suppose we let `Otext[(]t text[)]` stand for the odometer reading at time `t` — which is rarely zero in a real car. This function is also the solution of an initial value problem similar to the one that defines `stext[(]t text[)]`:

If we record the odometer readings at the beginning and end of a trip (at times `t=a` and `t=b`, respectively), then we compute the distance traveled as `Otext[(]btext[)]-Otext[(]atext[)]`.

The differential equations `s'text[(]t text[)]=vtext[(]t text[)]` and `O'text[(]t text[)]=vtext[(]t text[)]` are the same — each says that the derivative of the unknown function is the known function `vtext[(]t text[)].` [Think of `vtext[(]t text[)]` as the continuously recorded speedometer reading on the pen recorder.] Thus each initial value problem is asking for an antiderivative of `vtext[(]t text[)]`. Since the odometer reading at time `a` can be anything, we are really asking for any antiderivative of `vtext[(]t text[)]`. When we antidifferentiate, we find a family of functions of the form `Ftext[(]t text[)]+C`, where `F` is any function whose derivative is `v`, and `C` is any constant. In principle, every such function could be an odometer function, so the total distance traveled must be `[Ftext[(]btext[)]+C]-[Ftext[(]atext[)]+C]`, which is the same as `Ftext[(]btext[)]-Ftext[(]atext[)]`.

Notice that the answer obtained from this calculation does not depend on `C`. We know that from experience — no matter what the odometer says at the start of a trip, we can always use that number and the reading at the end of the trip to figure out how far we went.

The other answer we know to the distance-from-velocity question is the definite integral. That brings us to our first statement of an important part of the Fundamental Theorem:

where `F` is any function whose derivative is `v`.