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Real Analysis
book

Real Analysis

by V. Karunakaran
May 2024
Intermediate to advanced content levelIntermediate to advanced
585 pages
15h 38m
English
Pearson India
Content preview from Real Analysis
“real: chapter_08” 2011/5/22 23:34 page 12 #12
8-12 Real Analysis
to expect that this sum
n
i=1
f (t
i
)(x
i
x
i1
) should also approximate
b
a
f (x)dx to any desired accuracy if f R[a, b]. In the following we
shall show that this is the case when the partitions are chosen judi-
ciously. Further we will see later that proper choices of t
i
∈[x
i1
, x
i
]
will be very useful in determining
b
a
f (x)dx. The following example
illustrates this.
Example 8.3.9 Let f (x) = x
2
. Consider g(x) =
x
3
3
for which g
(x) =
f (x) for x ∈[0, 1]. We have already seen that g
R[a, b]. Let P =
{x
0
, x
1
, x
2
, ..., x
n
} be any partition of [0, 1]. The Mean Value Theorem
gives t
i
∈[ ...
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Publisher Resources

ISBN: 9781299447561Publisher Website