Calculus: Analytic Geometry and Calculus, with Vectors

4.2 Riemann sums and integrals 223

14 Assuming that the integrals exist, show that

(1)

f hf(x) dx=

fohf(-x) dx.

Remark: This innocent formula and a result of the next section enable us to pro-

duce the better formula

(2) f

hh

f (x) dx = f

oh

f (x) dx +

Joh f(x) dx

= fhU(-x) +f(x)jdx.

h

This gives the very useful fact thatf_hf(x) dx = 0 when f is an odd function,

that is, f(-x) = -f(x), and that

f

hhf(x)

dx = 2 ffh f(x) dx

when f is an even function, that is, f(-x) = f(x).

15 Remark: This remark is designed to indicate that mathematics is a lively

subject in which even good ideas can be modified in various ways, and that there

are integrals of many different types. We can be irked by the fact that the Rie-

mann integral

fogAX) dx

does not exist when f is the function for which f(x) = 1 when 0 < x < 1 and
f(x) = 2 when 1 < x <- 2. The difficulty is that f(1) is undefined and that
Ef(tk) Atk is undefined when tk = 1 for some k. If, however, we extend the
definition off by setting f(1) = 75, then the new extended function is Riemann
integrable over the interval 0 5 x S 2. We cannot reasonably undertake to
remove this irksome situation by changing the definition of the Riemann integral,
because changing basic definitions destroys our means for communication of
information. We can, however, introduce new types of integrals. We can, for
example, use the letter F to make us think of a finite set and produce the following
definition. A function f is Riemann-F integrable over a < x <_ b if there is a
finite set F such that f is defined at all points of the interval a < x < b except
at the points of F and, moreover, there is a number I such that to each e > 0
there corresponds a number S > 0 such that IRS - 11 < e whenever RS is a
Riemann sum for which IPI < 3 and the points tk are all different from points
of F. This definition does not require f to be defined everywhere over a 5 x 5 b
and it removes the irritation. Still another definition can be constructed by
making similar use of the letter C to make us think of a countable set of points,
this being either a finite set or a set whose elements can be placed in one-to-one
correspondence with the set 1, 2, 3, of positive integers. A more sophisti-
cated definition makes use of the letter N to make us think of a null set, this being
a set having Lebesgue measure 0. As has been remarked, there are many kinds
of integrals. Mathematicians who use integrals without knowing which ones
they are using are comparable to chemists who use chemicals without knowing
which ones they are using.