lim [U(f, Pn) – L(f, Pn)] = 0, and in this case Sf = lim,+. U (f, Pn) = lim,-+ L(f, Pn). (b) For each n, let Pn be the partition of [0, 1] into n equal subintervals. Find formulas for U(f, Pn) and L(f, Pn) if f(x) = x. The formula 1+ 2+ 3+ .·+n = n(n + 1)/2 will be useful. ... (c) Use the sequential criterion for integrability from (a) to show directly that f(x) = x is integrable on [0, 1] and compute f, f.

lim [U(f, Pn) – L(f, Pn)] = 0, and in this case Sf = lim,+. U (f, Pn) = lim,-+ L(f, Pn). (b) For each n, let Pn be the partition of [0, 1] into n equal subintervals. Find formulas for U(f, Pn) and L(f, Pn) if f(x) = x. The formula 1+ 2+ 3+ .·+n = n(n + 1)/2 will be useful. ... (c) Use the sequential criterion for integrability from (a) to show directly that f(x) = x is integrable on [0, 1] and compute f, f.

Name: (a) Prove that a bounded function f is integrable on [a, b] if and onl
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Description: (a) Prove that a bounded function f is integrable on [a, b] if and only if there exists a sequence of partitions (Pn)∞n=1 satisfying lim [U(f, Pn) – L(f, Pn)] = 0,and in this caseSf = lim,+. U (f, Pn) = lim,-+ L(f, Pn).(b) For each n, let Pn be the p

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Sequence and Series

A sequence is defined as a collection of things. Series is defined to sum the things one by one in the sequence. It was invented by German mathematician Carl Friedrich Gauss.

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Question

(a) Prove that a bounded function f is integrable on [a, b] if and only if there exists a sequence of partitions (Pn)∞n=1 satisfying

lim [U(f, Pn) – L(f, Pn)] = 0,
and in this case
Sf = lim,+. U (f, Pn) = lim,-+ L(f, Pn).
(b) For each n, let Pn be the partition of [0, 1] into n equal subintervals. Find
formulas for U(f, Pn) and L(f, Pn) if f(x) = x. The formula 1+ 2+ 3+
.·+n = n(n + 1)/2 will be useful.
...
(c) Use the sequential criterion for integrability from (a) to show directly that
f(x) = x is integrable on [0, 1] and compute f, f.

With differentiation, one of the major concepts of calculus. Integration involves the calculation of an integral, which is useful to find many quantities such as areas, volumes, and displacement.

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