4.4 Areas and integrals 237It is possible to describe complicated rules for constructing sets S for
which no such number ISI exists, and we say that such sets do not possess
area' (or are nonmeasurable). However, such sets are muchmore com-
plicated than those that appear in this book. This discussion of areas
will have served a purpose if it provides a reason for acceptance of the
fact that the theory of area is much more complicated than the theory of
Riemann integrals and that intuitive ideas about areas do not provide a
satisfactory basis for proofs of theorems about Riemann integrals. We
can, however, be reassured by the facts that many of the results of the
theory of area are thoroughly simple and that they are in complete agree-
ment with all of the results we shall obtain by use of Riemann integrals.
We shall not use Riemann integrals to obtain illusory information about
areas of sets that do not possess areas. More information about this
matter will appear in Section 5.7.
In what follows, we suppose that M, a, and b are constants and that
f is a function, Riemann integrable over a S x S b, for which 0 < f(x)
M when a S x < b. Let S be the set of points (x,y) for whicha x <_ b
and 0 < y < f (x). The set S may look more or less like the sets shown in
Figures 4.471 and 4.472. In each case we can describe S as the set of
a Aa b x
Figure 4.471points or part of the plane or region bounded on the left and right by the
graphs of the equations x = a and x = b and bounded below and above
by the graphs of the equations y = 0 and y = f(x). In case f is con-
tinuous and the graph of y = f(x) looks like that shown in Figure 4.472,
we can comfortably describe S as the region bounded by the graphs of
the four equations.$
t Newspapers and magazines keep us permanently aware of the fact that there are
inadequacies in old-fashioned intuitive physical theories of matter and that these intuitive
theories do not provide an adequate basis for modern physics. Since these newspapers and
magazines keep us quite generally uninformed about theories of areas and volumes, it may
be necessary to consult Appendix 2 at the end of this book to learn that there are bugs in
intuitive theories of areas and volumes.
I Of course climatologists who talk about areas of abundant rainfall, and philosophers
who talk about areas of scientific thought, could be expected to call S the area bounded by
the graphs. But in mathematics and perhaps even in climatology (we never know about
philosophy) an area is always a number and scientists do not, in their most brilliant
moments, call S an area.