Mechanics 1: Linear Kinematics and Calculus - 3D Math Primer for Graphics and Game Development - page 539

is not relevant. We use t to define the stopping point of the time range of
velocities to integrate.
Where does t 0 come from? It is an arbitrary starting point, reflecting
a degree of uncertainty (or freedom) very similar to the unknown (or irrel-
evant) starting position x 0 . We can pick t 0 to be whatever we want our
measurements to be relative to. The value of t 0 defines the point where
x(t) = 0. It's probably more precise to say that x(t) describes our relative
position. Relative to where? Wherever we were at time t 0 .
Now we're ready to clear up the sometimes confusing relationship be-
tween the definite and indefinite integral. The adjective “definite” in “def-
inite integral” comes from the fact that we have specified the limits of
integration. Because of this, the “answer” to a definite integral can be a
single number. When we evaluate a definite integral, such as

t end

v(t) dt,

t start

the t gets “integrated out” and does not appear in the result. The meaning
of the above is “the continuous summation of the velocity during the time
interval t start to t end .” It wouldn't make sense for the result to contain t—
which t would we be talking about? Thus, if all the other variables in v(t)
are known, and the limits t start and t end are known, we can boil down the
answer to a simple number. If, however, v(t) contains some other unknown
quantities (perhaps some variable density ρ), or the limits of integration
themselves are parameters, then the result will be function in terms of
those variables. In any case, in a definite integral the t will not be part of
the result. If you're a programmer, then you can think of the t as a “local
variable” to the definite integral.
An indefinite integral, on the other hand, since it is an antiderivative,
will have an “answer” that is function, not a single number. It is denoted
simply by dropping the limits of integration, such as

v(t) dt.

Again, we stress that while this may look very similar to the notation used
to denote a definite integral, its meaning is actually quite different. The
result of evaluating this integral should not be a number, but an antideriva-
tive of v(t); that is, we should get a function of t. Furthermore, a proper
result will have some arbitrary constant added, known as the constant of in-
tegration, which reminds us that there is a whole family of functions whose
derivative is v(t). Thus, the meaning of the indefinite integral above is
“some function that expresses the continuous summation of the velocity as
a function of time, from some unknown starting point.” We have been de-
noting this constant offset as x 0 , but in a more general setting it is typically