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2.1 The Area Under a Power Function
Let be a positive number, let be a positive number,
and let be the set of points in
such that and
. In this section we will
begin an investigation of the area of .
Our discussion will not apply to negative values of ,
since we make frequent use of the fact that for all non-negative
numbers and
Also
is not defined when is negative.
The figures for the argument given below are for the case ,
but you should observe
that the proof does not depend on the pictures.
Let be a positive integer, and for
, let
.
Then
for , so the points
divide
the interval into equal subintervals. For , let
If
, then
for some index , and
, so
Hence we have
and thus
|
(2.1) |
If , then
and
so
. Hence,
for all , and
hence
so that
|
(2.2) |
Now
and
Since the boxes intersect only along their boundaries, we have
and similarly
Thus it follows from equations (2.1) and (2.2) that
|
|
|
(2.4) |
The geometrical question of finding the area of has led us
to the numerical problem of finding the sum
We will study this problem in the next section.
2.5
Definition (Circumscribed box.)
Let
be the smallest box containing
. i.e.
Notice that
. Thus equation
(2.4) can be written as
|
(2.6) |
Observe that the outside terms in (2.6) do not depend on .
Now we will specialize to the case when .
A direct calculation shows that
There is a simple (?) formula for
, but it is not
particularly easy to guess this formula on the basis of these calculations.
With
the help of my computer, I checked that
Also
Thus by taking in equation (2.6) we see that
On the basis of the computer evidence it is very tempting to guess that
Next: 2.2 Some Summation Formulas
Up: 2. Some Area Calculations
Previous: 2. Some Area Calculations
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Ray Mayer
2007-09-07