1.2 Integration on the real line
We wish to define the Riemann integral of a continuous real function in such a way that if
is positive on some interval
, then the integral of
from
to
is the area under the graph of
between
and
. This is illustrated by the shaded part of Figure 8. To do this, we first split the interval
into a collection of subintervals called a partition.




Definitions
A partition of the interval
is a finite collection of subintervals of
,

for which

The length of the subinterval is
.
We use to denote the maximum length of all the subintervals, so

Given a partition of
, we can approximate the area under the graph of
between
and
by constructing a sequence of rectangles, as shown in Figure 9.

Here the th rectangle has base
and height
(so the top-right corner of the rectangle touches the curve). The area of the rectangle is
(see Figure 10). Note that we could equally have chosen the rectangle to be of height
for any point
in
, and the theory would still work. This is because, for a continuous function
, the difference between one set of choices of values for
,
, and another disappears when we take limits of partitions. We have chosen
merely for convenience.



Summing the areas of all the rectangles gives an approximation to the area under the graph. This sum is called the Riemann sum for , with respect to this particular partition. (You may have seen upper Riemann sum and lower Riemann sum defined slightly differently elsewhere.)
Definition
The Riemann sum for with respect to the partition

is the sum

We now calculate the Riemann sum for a particular choice of function and partition, and then ask you to do the same for a second function.
Example 1
Let , where
. Show that for

we have

and determine .
Solution
Each of the subintervals of
has length
. Therefore

Using the identity

we obtain

as required.
Finally, since is a basic null sequence, we see that

Now try the following exercise, making use of the identity

Exercise 1
Let , where
. Show that for

we have

and determine .
Answer
Each of the subintervals of
has length
. Therefore

as required.
Since is a basic null sequence, we see that

The Riemann sums of Example 1 approximate the area under the graph of
between
and
. The approximation improves as
increases, and we expect the limiting value
to actually be the area under the graph. However, to be sure that this limit gives us a sensible value, we should check that
for any sequence
of partitions of
such that
. The following important theorem, for which we omit the proof, provides this check.
Theorem 1
Let be a continuous function. Then there is a real number
such that

for any sequence of partitions of
such that
.
We can now define the Riemann integral of a continuous function.
Definition
Let be a continuous function, where
. The value
determined by Theorem 1 is called the Riemann integral of
over
, and it is denoted by

The theorem tells us that to calculate the Riemann integral of over
, we can make any choice of partitions
for which
and calculate
. Thus the calculation of Example 1 really does demonstrate that

We define the Riemann integral when
, as follows.
Definitions
Let be a continuous real function.
If , and
is contained in the domain of
, then we define

Also, for values of in the domain of
, we define

As we have discussed, for a continuous real function that takes only positive values on
, where
, the Riemann integral

measures the area under the graph of between
and
. If we no longer require
to be positive, then the integral still has a geometric meaning: it measures the signed area of the set between the curve
, the
-axis and the vertical lines
and
, where we count parts of the set above the
-axis as having positive area, and parts of the set below the
-axis as having negative area, as illustrated in Figure 11.




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