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The following argument is due to Pierre de Fermat (1601-1665) [19, pages
219-222]. 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
![\begin{displaymath}
S_a\subset\bigcup_{j=1}^n O_j
\end{displaymath}](img396.gif) |
(2.32) |
so
![\begin{displaymath}
\mbox{\rm area}(S_a)\leq\mbox{\rm area}(\displaystyle {\bigcup_{j=1}^n}O_j)=b \left(1-{1\over b^{n}}\right).
\end{displaymath}](img397.gif) |
(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
![\begin{displaymath}
a^{-{1\over n}}(1 - a^{-1}) \leq \mbox{\rm area}(S_a) \leq a^{1\over n}(1-a^{-1}).
\end{displaymath}](img409.gif) |
(2.34) |
2.35
Exercise.
What do you think the area of
![$S_a$](img381.gif)
should be? Explain your answer. If
you
have no ideas, take
![$a=2$](img410.gif)
in (
2.34), take large values of
![$n$](img9.gif)
,
and by using a calculator, estimate
![$\mbox{\rm area}(S_a)$](img411.gif)
to three or
four decimal places of
accuracy.
2.36
Exercise.
A
Let
![$a$](img31.gif)
be a real number with
![$0<a<1$](img412.gif)
, and let
![$N$](img413.gif)
be a positive integer.
Then
Let
![$T_a$](img310.gif)
be the set of points
![$(x,y)$](img94.gif)
such that
![$a\leq x\leq 1$](img415.gif)
and
![$0\leq
y\leq\displaystyle { {1\over {x^2}}}$](img416.gif)
. Draw a sketch of
![$T_a$](img310.gif)
, and show that
The calculation of
![$\mbox{\rm area}(T_a)$](img418.gif)
is very similar to the calculation of
![$\mbox{\rm area}(S_a)$](img411.gif)
.
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
![$S_a^r$](img176.gif)
is for an arbitrary positive integer
![$r$](img175.gif)
.
Explain the basis for your guess. ( The correct formula for
![$\mbox{\rm area}(S_a^r)$](img419.gif)
for positive integers
![$r$](img175.gif)
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
2007-09-07