Here we will use a physics setting to see why the fundamental theorems of calculus could be true.

Let *f* (*t*)*a*,*b*].

Imagine that you are driving along a highway and at time
*t* = *a*,*f* (*t*),*t* = *b*.*F*(*x*)*t* = *x**F* here is called *r* in that note.

When you drive for *t* hours at a constant speed *v*, you cover
*d* = *vt**f* is a "nice"
function. This means that if, at some time *t*, you look at it over
some really tiny time interval *dt*, the velocity almost does not
change. Therefore, the change of your position *ds* during the time
segment *dt* is *f* (*t*)*dt*. The
total change in position (displacement) is obtained by summing all the tiny
changes over all possible time segments *dt* between
*t* = *a**t* = *x*:

Now we also see why we cannot use *x* as a variable in the integral when
we use it as a limit of this integral. When we fix one such *x*, we
still want to be able to move along time as we add up the small velocity
contributions, therefore we need another variable.

In any case, we just saw that the position *F*(*x*)*F* ′ = *f**F* is an
antiderivative of *f*, exactly as claimed by TFC 1.

Now imagine that at time *t* = *a*,*F* to record our position along the highway based on mileposts, in
particular, *F*(*a*)*F* is an antiderivative of the
velocity *f*. The displacement (distance we drove) between times
*t* = *a**t* = *b**F*, but also summing up the velocities as above, and
we get

This is exactly the Newton-Leibniz formula.

This equality is sometimes stated in this way:

In our interpretation it means the following: To find your position at time
*b*, you start with the position at time *a* and add all the
displacements corresponding to your instantaneous velocity between times
*a* and *b*.