360 Chapter 7 • The Riemann Integral
Proof. Exercise 5. •EXERCISE SET 7.1- Prove Theorem 7 .1. 2.
- Prove the remaining part of Theorem 7.1.3 (a), and (b)-(d).
- Prove Theorem 7.1.4.
- Prove that in Theorem 7.1.5 (b) and ( c), "<" can be replaced by "~".
- Prove Theorem 7.1.6.
7 .2 The Riemann Integral Defined
The "definite" integral l: f(x)dx or, more simply, l: f is quite familiar to
you from your elementary calculus course. In this section we give it a rigorous
definition. In honor of the mathematician Bernard Riemann (1826-1866) who
put this notion on a rigorous foundation, we call it the Riemann integral.
There are other definitions of l: f, attributed to other mathematicians and
useful for other, usually more advanced, purposes. However, Riemann's integral
is by far the most universally recognized and used at this level. We emphasize
that we are defining the definite integral, not the indefinite integral. The definite
integral is a fundamentally significant concept, existing independently of any
connection with derivatives. The "indefinite" integral l f(x)dx, on the other
hand, depends upon the notion of derivative for its definition. It represents
the general "antiderivative" off, and is often used as a tool in calculating the
definite integral or in expressing formal solutions of differential equations.
Assumption: Throughout this section and the next, unless otherwise spec-
ified, we shall assume that any function f to be "integrated" is defined and
bounded on a compact interval [a, b], with a < b. Any attempt to define
l: f where either (a, b) is not bounded or f is not bounded on [a, b] leads to
what we call "improper integrals,'' which we do not discuss until Section 7.8.
Definition 7.2.1 A partition of the interval [a, b] is a subset P =
{xo,X1,X2, · · · ,xn} s;;; [a,b], such that a= xo < X1 < x2 < · · · < Xn = b. (See
Figure 7.1 (a).)