1.2. Basic Properties of an Integral 3
II. Monotonicity: If the functions f, g V satisfy f(x)
g(x) for all x and a, b I satisfy a b, then
b
a
f(x) dx
b
a
g(x) dx.
In particular, if f(x) 0 for all x and a b, then
b
a
f(x) dx 0.
III. Additivity: For any function f V, and any a, b, c I,
c
a
f(x) dx =
b
a
f(x) dx +
c
b
f(x) dx.
In particular, we allow a, b and c to occur in any order on the
line and we note that two easy consequences of additivity
are
a
a
f(x) dx = 0 and
b
a
f(x) dx =
a
b
f(x) dx.
IV. Constant functions: The integral of a constant function
f(x) = C should be given by
b
a
C dx = C(b a).
If C 0 and a b, this just says the integral of f is the
area of the rectangle under its graph.
V. Finite sets don’t matter: If f and g are functions in V
with f(x) = g(x) for all x except possibly a finite set, then
for all a, b I,
b
a
f(x) dx =
b
a
g(x) dx.
Properties III, IV and V are not valid for all mathematically in-
teresting theories of integration. Nevertheless, they hold for all the
integrals we will consider, so we include them in our list of basic prop-
erties. It is important to note that these are assumptions, however,
and there are many mathematically interesting theories where they
do not hold.
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