4 Integrals
4.1 Indefinite integrals There are about as many different types of
integrals in mathematics as there are elements in chemistry, but only a
few of them occur in first courses in calculus. This chapter introduces
basic ideas about two kinds of integrals. These ideas may not be coming
too soon to meet the needs of students taking other courses in which
mathematics appears. In this section, and in some other places where
the deviation from complete linguistic rectitude does not create deceptive
statements, we sometimes refer to "the function f(x)" or to "the function
having values f(x)" instead of to the function f which, for each x in some
interval, has the value f(x).
It is very often true that we have a given function f(x) and we are
interested in those functions F(x) or y or y(x) (if any) for which
(4.11) F(x) = f(x) or dz = f(x)when x lies in some interval. Before discussing this situation, we intro-
duce notation that is universally used. In case F(x) or y is a function for
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