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If is a monotonic function from an interval to
,
then we
have shown that for every sequence of partitions on such that
, and every sequence such that
for all
is a sample for ,
we have
8.1
Definition (Integral.)
Let
be a bounded function from an interval
to
. We say that
is
integrable on
if there is a number
such that for every sequence of partitions
on
such that
, and every sequence
where
is a sample for
If
is integrable on
then the number
just described is denoted
by
and is called `` the integral from
to
of
.''
Notice that by our definition an integrable function is necessarily
bounded.
The definition just given is essentially due to Bernhard Riemann(1826-1866),
and
first appeared around 1860[39, pages 239 ff]. The symbol
was introduced by Leibniz sometime around 1675[15, vol 2, p242]. The symbol is a form of the letter s, standing for sum
(in Latin as well as in English.) The practice of attaching the
limits and
to the integral sign was introduced by Joseph Fourier
around 1820. Before this time the limits were usually indicated
by words.
We can now restate theorems 7.6 and 7.15 as follows:
In general integrable functions may take negative as well as positive values
and in
these cases
does not represent an area.
The next theorem shows that monotonic functions are integrable even if
they take on negative values.
8.4
Example (Monotonic functions are integrable II.)
Let
be a
monotonic function from an interval
to
. Let
be a non-positive number such that
for all
. Let
.
Then
is a monotonic function from
to
.
Hence by theorem
7.6,
is integrable on
and
. Now let
be a sequence of partitions of
such that
, and let
be a sequence such that for each
in
,
is a sample for
. Then
|
(8.5) |
If
and
then
Thus by (
8.5)
If we use the fact that
, and then use the sum theorem
for limits of sequences, we get
It follows from the definition of integrable functions that
is integrable
on
and
Thus in figure b,
represents the shaded area with the area of
the
thick box subtracted from it, which is the same as the area of the region
marked ``" in figures c and d, with the area of the region marked
``"
subtracted from it.
The figure represents a geometric interpretation for a Riemann sum. In the
figure
is the area of
and
is the negative of the area of
In general you should think of
as the difference
where
and
8.7
Exercise.
Below is the graph of a function
.
By looking at the graph of
estimate the following integrals.
(No explanation is necessary.)
Graph of
- a)
-
.
- b)
-
.
- c)
-
.
Next: 8.2 Properties of the
Up: 8. Integrable Functions
Previous: 8. Integrable Functions
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Ray Mayer
2007-09-07