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.