Define a partition of an interval. Write any three different examples of
partitions of [0,1].
(ii). Let a real valued function π be defined and bounded over [π, π]. Prove that π is
Riemann integrable over [π, π] if for each π > 0 there is a partition π such that
π(π, π) β πΏ(π, π) < π
(iii). Show that π(π₯) = π₯2
is Riemann integrable over [0,2].
Ans:-
Partition of a Set is defined as "A collection of disjoint subsets of a given set. The union of the subsets must equal the entire original set".
three different examples of partitions of are
Let f be bounded on . Then f is Riemann integrable if and only if for every Ο΅ there is a partition on such that:
Since
there should exist two partitions such that:
and
The argument is almost identical for (except flipping some inequalities)
Hence the function is Riemann integrable
For Reimann integrable
So We have this expression equal to
Here
Hence is Riemann integrable over
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