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CHAPTER 4
The Integral
4.0 Introduction
Many processes, both in mathematics and in nature, involve addition.You are famil-
iar with the discrete process of addition, in which you add finitely many numbers
to obtain a sum or aggregate. But there are important instances in which we wish
to add infinitely many terms. One important example is in the calculation of area—
especially the area of an unusual (non-rectilinear) shape. A standard strategy is to
approximate the desired area by the sum of small, thin rectangular regions (whose
areas are easy to calculate). A second example is the calculation of work, in which
we think of the work performed over an interval or curve as the aggregate of small
increments of work performed over very short intervals. We need a mathematical
formalism for making such summation processes natural and comfortable. Thus we
will develop the concept of the integral.
4.1 Antiderivatives and Indefinite
Integrals
4.1.1 THE CONCEPT OF ANTIDERIVATIVE
Let f be a given function. We have already seen in the theory of falling bodies
(Section 3.4) that it can be useful to find a function F such that F = f . We call
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