Graphics Reference
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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
 
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