1549901369-Elements_of_Real_Analysis__Denlinger_

364 Chapter 7 • The Riemann Integral

Finally, we are able to define the definite integral.

Definition 7.2.8 ( Darboux's Definition of l: !) A function f defined and

bounded on [a, b] is integrable on [a, b] if l: f = l: f. In this case, the common

value of l: f and l: f is called the (definite) Riemann integral off over [a, b],

and is denoted simply l: f.

Note on Notation: We have not used the common notation l: f(x)dx, famil-

iar to you from elementary calculus, because in defining the definite integral

the symbols x and dx play no role. The notation l: f correctly indicates that

all we need are a function and an interval. We gain simplicity by omitting x

and dx. However, in concrete examples we will often find it helpful to use the

more familiar notation, l: f(x)dx.

Example 7.2.9 A constant function f(x) = c is integrable over [a, b], and

l:f=c(b-a).

Proof. For every partition P of [a, b], and for every i = 1, 2, · · · , n,

n n n

i=l i=l i=l

and hence, l: f = c(b -a). Similarly, for every partition P of [a, b],

n n n

i=l i=l i=l

and hence l: f = c(b-a). Thus, l: f = l: f = c(b-a), from which the desired

conclusion follows. 0 -

Example 7.2.10 (A Nonintegrable Function) The Dirichlet function^4

f : m .1r<>. ---> m .1r<>. given. b y !( x ) = { 1 if x is rational, } is not integra b le on any
0 if x is irrational
closed interval [a, b], where a < b.

4. See Example 5.1.11.