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7.1
Theorem.
Let be a monotonic function from the interval to
. Let be a sequence of partitions of such
that
, and let
Then
and
(The notation here is the same as in theorem 5.40 and exercise
5.47.
Proof: We noted in theorem 5.40 and exercise 5.47 that
|
(7.2) |
Since
we conclude from the squeezing rule that
|
(7.3) |
We also have by (5.43) that
so that
By (7.3) and the squeezing rule
and hence
Also,
7.4
Definition (Riemann sum.)
Let
be a partition for an
interval
. A
sample for
is a finite sequence
of numbers such that
for
. If
is
a function from
to
, and
is a partition for
and
is
a
sample for
, we define
and
we call
a
Riemann sum for
and
.
We will sometimes write
instead of
.
7.5
Example.
If
is an increasing function from
to
, and
is a partition of
, and
, then
If
, then
If
then
is some number between
and
.
Proof: We will consider the case where is increasing. The case where
is
decreasing is similar.
For each partition
and sample
,
we have for
Hence
i.e.,
By theorem 7.1 we have
and
so by the squeezing rule,
Next: 7.2 Calculation of Area
Up: 7. Still More Area
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Ray Mayer
2007-09-07