1.3 Properties of the Riemann integral
In practice we do not usually calculate integrals by looking at partitions, but instead use a powerful theorem known as the Fundamental Theorem of Calculus, which allows us to think of integration and differentiation as inverse processes.
To state the theorem, we need the notion of a primitive of a continuous real function ; this is a real function
that is differentiable on
with derivative equal to
, that is, the function
satisfies
, for all
. A primitive of a function is not unique, because if
is a primitive of
, then so is the function with rule
, for any constant
.
Theorem 2 Fundamental Theorem of Calculus
Let be a continuous function. If
is a primitive of
, then the Riemann integral of
over
exists and is given by

We denote by
.
For example, a primitive of is
, so

which agrees with our earlier calculation using Riemann sums.
The Riemann integral has a number of useful properties.
Theorem 3 Properties of the Riemann integral
Let and
be real functions that are continuous on the interval
.
Sum Rule
.
Multiple Rule
, for
.
Additivity Rule
Substitution Rule If
is differentiable on
and its derivative
is continuous on
, and if
is continuous on
, then
Integration by Parts If
and
are differentiable on
and their derivatives
and
are continuous on
, then
Monotonicity Inequality If
for each
, then
Modulus Inequality
.
The first five properties are probably familiar to you and we have stated them only for reference. The last two inequalities may be less familiar. The Monotonicity Inequality, illustrated in Figure 12, states that if you replace by a greater function
, then the integral increases.

The Modulus Inequality, illustrated in Figure 13, says that the modulus of the integral of over
(a non-negative number) is less than or equal to the integral of the modulus of
over
(another non-negative number). If
is positive, then these two numbers are equal, but if
takes negative values, then at least part of the signed area between
and the
-axis is negative, so the first number is less than the second.

Exercise 2
Use the Monotonicity Inequality and the fact that

to estimate from above and below.
Answer
Since

it follows from the Monotonicity Inequality that

Hence

that is,

Since and
, we see that

(In fact, to two decimal places.)
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