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Index
The following argument is due to Pierre de Fermat (16011665) [19, pages
219222]. Later we will use Fermat's method to find the area under
the
curve for all in
.
Let be a real number with , and let be the set of points
in
such that and
. I
want to
find the area of .
Let be a positive integer. Note that since , we have
Let be the box
Thus the upper left corner of lies on the curve
.
To simplify the notation, I will write
Then
and
Hence
Observe that we have here a finite geometric series, so
Now

(2.32) 
so

(2.33) 
Let be the box
so that the upper right corner of lies on the curve
and
lies
underneath the curve
. Then
Hence,
Since
we have
i.e.,
By combining this result with (2.33), we get
Since
, we can rewrite this as

(2.34) 
2.35
Exercise.
What do you think the area of
should be? Explain your answer. If
you
have no ideas, take
in (
2.34), take large values of
,
and by using a calculator, estimate
to three or
four decimal places of
accuracy.
2.36
Exercise.
A
Let
be a real number with
, and let
be a positive integer.
Then
Let
be the set of points
such that
and
. Draw a sketch of
, and show that
The calculation of
is very similar to the calculation of
.
What do you think the area of should be?
2.37
Exercise.
Using the inequalities (
2.6), and the
results of Bernoulli's table in section
2.2, try to guess what
the area of
is for an arbitrary positive integer
.
Explain the basis for your guess. ( The correct formula for
for positive integers
was stated by
Bonaventura Cavalieri in 1647[
6, 122 ff].
Cavalieri also found
a method for computing general positive integer
power sums.)
Next: 2.6 Area of a
Up: 2. Some Area Calculations
Previous: 2.4 Finite Geometric Series
Index
Ray Mayer
20070907