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(iv) Provethat (
C
) is a monotonic decreasing sequence.
Deduce that this sequence is convergent to a limit
C
, say.
(v) Provethat
C
I
((
b
a
)/
n
)(
f
(
b
)
f
(
a
)), and deduce that
) is a null sequence.
(vi) Use qn 3.54(v), the difference rule, to show that
C
(
C
I
I
, so that
we have found a measure of the area bounded by the graph of
y
f
(
x
), thelins
x
a
,
x
b
and the
x
-axis.
8 In qn 7, had
f
been monotonic decreasing, in what way would the
argument have been affected?
9 (
Gregory of St Vincent
, after
Toeplitz
, 1963) A famous example of a
continuous function which is monotonic decreasing is the function
given by
f
(
x
)
1/
x
for positivevalus of
x
. From qn 8, thearea
bounded by this graph, the
x
-axis and thelins
x
a
and
x
b
,
for positive
a
and
b
, is well defined, being the limit of the sum of
the areas of inscribed rectangles.
(i) Show that if
n
rectangles with bases of equal width are
inscribed under this graph on the interval [1,
a
], and that if
n
rectangles with bases of equal width are inscribed under this
graph on theintrval [
b
,
ba
], then the area of the two sets of
n
rectangles areequal.
(ii) Deduce from qn 8 that the area under the graph on [1,
a
]is
equal to the area under the graph on [
b
,
ba
].
(iii) Deduce further that the area under the graph on [1,
ab
]is
equal to the sum of the areas under the graph on [1,
a
] and
[1,
b
], the essential property of the logarithm function.
(iv) Interpret this result in terms of the limits of qn 6.
The definite integral
It is now appropriate to recognise that the argument in qn 7 did
not require the function
f
to have positive values except for the purpose
of using theword 'area'. The inequalities of qn 7(i) are independent of
any notion of area.
In qn 7(ii) wecan define
l
n
f
(
i
1)
l
n
l
n
f
il
n
I
a
and
C
a
,
without reference to area, and we obtain convergent sequences as before
in qn 7(iii) and 7(iv), with equal limits, as in qn 7(vi). The common limit
is called the
integral
of thefunction
f
on [
a
,
b
], and is denoted by
f
.
Thesymbol
is a largesquashed
S
(for sum).
