Mechanics 1: Linear Kinematics and Calculus - 3D Math Primer for Graphics and Game Development - page 538

Fundamental Theorem of Calculus, Part 2

Let F(t) be defined by

t

F(t) = f(u) du. (11.27)

t 0

Then the derivative of F(t) is given by

′

F (t) = f(t).

It can take some effort to decipher this terse elegance, so let's restate it
in English. We start with a given function f. We then form a new func-
tion F, whose value is determined by taking the definite integral of f from
any arbitrary starting point t 0 , and an ending point t. Note that the ar-
gument to F is used to define when to stop the integration of f. The
variable u is a notational dummy variable of integration; it is not seen
outside of the integral. The second fundamental theorem of calculus says
that if we take the derivative of this new function F, the result is our orig-
inal function f. In this sense, integration and differentiation are inverse
operations.
It can be di cult to grasp the reason why t ends up in what may seem
to be an odd location, defining the upper limit of the integration, but that
is the essential point. The second theorem is saying that a function defined
as an integral such as Equation (11.27) will grow at a rate determined by
the integrand. If we adjust the upper limit of integration a tiny bit, the
change in the result of the overall sum will be proportional to the value
of integrand. Thinking of an integral as calculating an area, the upper
limit of integration, t, determines the right-hand boundary. If we push this
boundary to the right a bit, the increase in the amount of area will depend
on the height of the function at t.
Let's rewrite the theorem using notation particular to displacements
and velocities:

t

x(t) = v(u) du,

t 0

′

x (t) = v(t).

Now we see that, to define the displacement x(t) in terms of v(t), there's
really only one logical place we could put t. The velocity before t is relevant
to the displacement that had occurred by time t, and the history after t