Graphics Reference
In-Depth Information
Now, cover
A
with rectangles with bases (0, 0)(
a
/
n
, 0),
(
a
/
n
, 0)(2
a
/
n
, 0), . . ., ((
n
1)
a
/
n
, 0)(
a
, 0), to show that
a
n
A
n
a
(1
2
...
)
(1
1/
n
)(1
1/2
n
).
Deduce that
A
a
is theonly valuewhich satisfis the
inequalities
a
(1
1/
n
)(1
1/2
n
)
A
a
(1
1/
n
)(1
1/2
n
)
for all values of
n
.
2 Let
B
be the area bounded by the curve
y
x
, the
x
-axis and the
line
x
a
, where we take
a
to be positive. Using the method of qn
1, and appealing to qn 1.3(iii), prove that, for all values of the
positive integer
n
,
a
B
a
(1
1/
n
)
(1
1/
n
)
.
Deduce that
B
a
is the only value which satisfies all of these
inequalities.
Using this method to find the area bounded by the curve
y
x
,
the
x
-axis and theline
x
a
depends upon knowing a formula for the
sum
r
, which we may not have to hand. Faced with this problem,
Fermat devised a new approach, which enabled him to find areas under
curves of the form
y
1/
x
, for rational
k
1 and curves of the form
y
x
for rational
k
.
3 (
Fermat
, 1658) Let
C
be the area between the curve
y
, the
x
-axis and theline
x
1. Let
r
1. By inscribing rectangles in
C
,
parallel to the
y
-axis, with bases (1, 0)(
r
, 0),
(
r
, 0)(
r
, 0), . . ., (
r
,0)(
r
, 0), . . ., provethat 1/
r
C
, and by
circumscribing rectangles with the same bases around
C
, provethat
C
r
. Since1/
r
C
r
for every real number
r
1, deduce that
C
1.
1/
x
4 Let
C
be the area between the curve
y
x
, the
x
-axis and theline
x
a
, where
k
is a positive integer and
a
is positive. Let 0
r
1.
By inscribing rectangles in
C
with bases (
a
,0)(
ar
, 0),
(
ar
, 0)(
ar
, 0), . . ., (
ar
,0)(
ar
, 0), . . ., provethat
r
a
r
C
,
1
r
r
...
and by circumscribing rectangles with the same bases around
C
,
provethat