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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
![\begin{displaymath}
0 \leq \alpha\Big( O_a^b(f,P_n)\Big)-\alpha\Big( I_a^b(f,P_n)\Big)\leq
\mu (P_n)\cdot\vert f(b)-f(a)\vert.
\end{displaymath}](img1857.gif) |
(7.2) |
Since
we conclude from the squeezing rule that
![\begin{displaymath}
\lim \left\{ \alpha\Big( O_a^b(f,P_n)\Big)-\alpha\Big(I_a^b(f,P_n)\Big)\right\}
=0.
\end{displaymath}](img1859.gif) |
(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
![$P=\{x_0,x_1,\cdots ,x_n\}$](img1169.gif)
be a partition for an
interval
![$[a,b]$](img1071.gif)
. A
sample for
![$P$](img550.gif)
is a finite sequence
![$S=\{s_1,s_2,\cdots
,s_n\}$](img1865.gif)
of numbers such that
![$s_i\in[x_{i-1},x_i]$](img1866.gif)
for
![$1\leq i\leq n$](img188.gif)
. If
![$f$](img676.gif)
is
a function from
![$[a,b]$](img1071.gif)
to
![$\mbox{{\bf R}}$](img153.gif)
, and
![$P$](img550.gif)
is a partition for
![$[a,b]$](img1071.gif)
and
![$S$](img49.gif)
is
a
sample for
![$P$](img550.gif)
, we define
and
we call
![$\sum (f,P,S)$](img1868.gif)
a
Riemann sum for
![$f,\;P$](img1869.gif)
and
![$S$](img49.gif)
.
We will sometimes write
![$\sum([f(t)],P,S)$](img1870.gif)
instead of
![$\sum (f,P,S)$](img1868.gif)
.
7.5
Example.
If
![$f$](img676.gif)
is an increasing function from
![$[a,b]$](img1071.gif)
to
![$\mbox{{\bf R}}_{\geq 0}$](img1168.gif)
, and
![$P=\{x_0,\cdots ,x_n\}$](img1142.gif)
is a partition of
![$[a,b]$](img1071.gif)
, and
![$S_l=\{x_0,\cdots
,x_{n-1}\}$](img1871.gif)
, then
If
![$S_r=\{x_1,x_2,\cdots ,x_n\}$](img1873.gif)
, then
If
![$\displaystyle {S_m=\{ {{x_0+x_1}\over 2},\cdots , {{x_{n-1}+x_n}\over 2}\}}$](img1875.gif)
then
is some number between
![$\alpha(I_a^b(f,P))$](img1394.gif)
and
![$\alpha(O_a^b(f,P))$](img1395.gif)
.
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
Previous: 7. Still More Area
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Ray Mayer
2007-09-07