Page 100 - Schaum's Outline of Theory and Problems of Advanced Calculus

CHAP. 5] INTEGRALS 91

Subdivide the interval a @ x @ b into n sub-intervals by means of the points x 1 ; x 2 ; ... ; x n 1 chosen
arbitrarily. In each of the new intervals ða; x 1 Þ; ðx 1 ; x 2 Þ; ... ; ðx n 1 ; bÞ choose points 1 ; 2 ; ... ; n
arbitrarily. Form the sum

f ð 1 Þðx 1 aÞþ f ð 2 Þðx 2 x 1 Þþ f ð 3 Þðx 3 x 2 Þþ þ f ð n Þðb x n 1 Þ ð1Þ
By writing x 0 ¼ a, x n ¼ b; and x k x k 1 ¼ x k , this can be written
n n
X X
f ð k Þ x k
f ð k Þðx k x k 1 Þ¼ ð2Þ
k¼1 k¼1
Geometrically, this sum represents the total area of all rectangles in the above figure.
We now let the number of subdivisions n increase in such a way that each x k ! 0. If as a result
the sum (1)or(2) approaches a limit which does not depend on the mode of subdivision, we denote this
limit by
ð b n
X
f ðxÞ dx ¼ lim f ð k Þ x k ð3Þ
a n!1 k¼1
This is called the definite integral of f ðxÞ between a and b.In this symbol f ðxÞ dx is called the integrand,
and ½a; bŠ is called the range of integration.We call a and b the limits of integration, a being the lower
limit of integration and b the upper limit.
The limit (3) exists whenever f ðxÞ is continuous (or piecewise continuous) in a @ x @ b (see Problem
5.31). When this limit exists we say that f is Riemann integrable or simply integrable in ½a; bŠ.
The definition of the definite integral as the limit of a sum was established by Cauchy around 1825.
It was named for Riemann because he made extensive use of it in this 1850 exposition of integration.
Geometrically the value of this definite integral represents the area bounded by the curve y ¼ f ðxÞ,
the x-axis and the ordinates at x ¼ a and x ¼ b only if f ðxÞ A 0. If f ðxÞ is sometimes positive and
sometimes negative, the definite integral represents the algebraic sum of the areas above and below the x-
axis, treating areas above the x-axis as positive and areas below the x-axis as negative.

MEASURE ZERO
A set of points on the x-axis is said to have measure zero if the sum of the lengths of intervals
enclosing all the points can be made arbitrary small (less than any given positive number ). We can
show (see Problem 5.6) that any countable set of points on the real axis has measure zero. In particular,
the set of rational numbers which is countable (see Problems 1.17 and 1.59, Chapter 1), has measure
zero.
An important theorem in the theory of Riemann integration is the following:

Theorem. If f ðxÞ is bounded in ½a; bŠ, then a necessary and sufficient condition for the existence of
Ð b f ðxÞ dx is that the set of discontinuities of f ðxÞ have measure zero.
a

PROPERTIES OF DEFINITE INTEGRALS
If f ðxÞ and gðxÞ are integrable in ½a; bŠ then
ð b ð b ð b
1. f f ðxÞ gðxÞg dx ¼ f ðxÞ dx gðxÞ dx
a a a
b b
ð ð
2. Af ðxÞ dx ¼ A f ðxÞ dx where A is any constant
a a