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written with a capital C. For example,
v(t) dt = x(t) + x
0
(displacement and velocity notation),
Constant of integration
f(t) dt = F(t) + C
(common abstract notation).
We do not need to write the limits of integration in an indefinite integral
because they are implicit. As we saw in the second part of the fundamental
theorem of calculus, the interpretation of an antiderivative in terms of a
definite integral is to use the argument of the antiderivative as the upper
limit of the range of integration. In other words, an indefinite integral is
simply a definite integral with implied limits of integration of the form in
Equation (11.27). The degree of freedom in Equation (11.27) connecting
the set of possible antiderivatives was captured by the unknown lower limit
of integration (t
0
). In an indefinite integral we don't write the limits of
integration, and instead the uncertainty is contained in the constant of
integration (x
0
or C). We can summarize this (written using both naming
schemes) by
t
The indefinite integral
v(t) dt =
v(u) du = x(t) + x
0
,
x
0
= −x(t
0
),
t
0
t
f(t) dt =
v(u) du = F(t) + C,
C = −F(t
0
).
t
0
11.7.3 Summary of Calculus
We have completed our main presentation of calculus in this topic, aside
from a few small bits that come up in later sections. Our goal has been
to take a reader with absolutely no knowledge of calculus to a point where
that reader understands the big picture of what derivatives and integrals
are used for. We have whizzed right past the many, many details and
techniques that arise in practical situations—these details fill up thousands
of pages in calculus textbooks.
Let's summarize the important points that you need to know about
calculus to fully utilize the remainder of this topic.
•
The basic purpose of a derivative is to measure a rate of change.
•
The derivative is defined by using a limit argument. We form an
approximation of the result, and then watch what happens as we take
better and better approximations in the limit as our error approaches
zero.
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