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Subdivide the interval a @ x @ b into n sub-intervals by means of the points x1 ; x2 ; . . . ; xn 1 chosen arbitrarily. In each of the new intervals a; x1 ; x1 ; x2 ; . . . ; xn 1 ; b choose points 1 ; 2 ; . . . ; n arbitrarily. Form the sum f 1 x1 a f 2 x2 x1 f 3 x3 x2 f n b xn 1 By writing x0 a, xn b; and xk xk 1 xk , this can be written
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Geometrically, this sum represents the total area of all rectangles in the above gure. We now let the number of subdivisions n increase in such a way that each xk ! 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 b n X f x dx lim f k xk 3
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This is called the de nite integral of f x between a and b. In this symbol f x dx is called the integrand, and a; b 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 x 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 a; b . The de nition of the de nite 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 de nite integral represents the area bounded by the curve y f x , the x-axis and the ordinates at x a and x b only if f x A 0. If f x is sometimes positive and sometimes negative, the de nite integral represents the algebraic sum of the areas above and below the xaxis, 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, 1), has measure zero. An important theorem in the theory of Riemann integration is the following: Theorem. If f x is bounded in a; b , then a necessary and su cient condition for the existence of b a f x dx is that the set of discontinuities of f x have measure zero.
PROPERTIES OF DEFINITE INTEGRALS If f x and g x are integrable in a; b then b 1.
f f x g x g dx
f x dx b
b g x dx
b 2.
A f x dx
f x dx
where A is any constant
92 b 3.
INTEGRALS
[CHAP. 5
f x dx
f x dx a
f x dx
provided f x is integrable in a; c and c; b .
b 4.
f x dx
f x dx
a 5.
f x dx 0
If in a @ x @ b, m @ f x @ M where m and M are constants, then m b a @ b
f x dx @ M b a
If in a @ x @ b, f x @ g x then b f x dx @
b g x dx
  b   b   j f x j dx  f x dx @   a a
if a < b
MEAN VALUE THEOREMS FOR INTEGRALS As in di erential calculus the mean value theorems listed below are existence theorems. The rst one generalizes the idea of nding an arithmetic mean (i.e., an average value of a given set of values) to a continuous function over an interval. The second mean value theorem is an extension of the rst one that de nes a weighted average of a continuous function. By analogy, consider determining the arithmetic mean (i.e., average value) of temperatures at noon for a given week. This question is resolved by recording the 7 temperatures, adding them, and dividing by 7. To generalize from the notion of arithmetic mean and ask for the average temperature for the week is much more complicated because the spectrum of temperatures is now continuous. However, it is reasonable to believe that there exists a time at which the average temperature takes place. The manner in which the integral can be employed to resolve the question is suggested by the following example. Let f be continuous on the closed interval a @ x @ b. Assume the function is represented by the correspondence y f x , with f x > 0. Insert points of equal subdivision, a x0 ; x1 ; . . . ; xn b. Then all xk xk xk 1 are equal and each can be designated by x. Observe that b a n x. Let k be the midpoint of the interval xk and f k the value of f there. Then the average of these functional values is
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