Fermat’s Determination of the Area under
x
n
over [0,b]
Consider the function
f
(
x
)
=
x
n
over the interval
[0,
b
]
.
Let
r
be any number
such that
0
<
r
<
1
.
Fermat got the idea to approximate the area under the curve with an
infinite number of rectangles as in the graph below.
Only the rightmost six rectangles are
shown on the graph.
The areas of the rightmost three of the rectangles are .
The area of the
k
th
rectangle can be seen to have the following formula.
(
r
k
!
b
)
n
!
r
k
!
b
!
(1
"
r
)
=
b
n
+
1
!
(1
"
r
)
!
r
(
n
+
1)
!
k
So the total area of the rectangles is given by the following.
b
n
+
1
!
(1
"
r
)
!
r
(
n
+
1)
!
k
k
=
0
#
$
Using the formula for the Geometric Series we get the following.
b
n
+
1
!
(1
"
r
)
1
"
r
n
+
1
=
b
n
+
1
1
"
r
n
+
1
1
"
r
=
b
n
+
1
1
+
r
+
r
2
+
!
+
r
n
Then Fermat took the limit of this expression as
r
!
1
from below.
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 Spring '07
 JURY
 Calculus, Calculus Gems

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