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.

Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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(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.
Transcribed Image Text: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.
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