(b) Prove that if f,g are integrable on S, then so is f + g, and +9) = /. f + g. S S (First use part (a) to prove that f s(f +g) < Ssf + S sg and ſ ,(f+9) > Lgƒ + L,9-)
(b) Prove that if f,g are integrable on S, then so is f + g, and +9) = /. f + g. S S (First use part (a) to prove that f s(f +g) < Ssf + S sg and ſ ,(f+9) > Lgƒ + L,9-)
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
Please solve 2(b)
![#2 Let S be a closed n-cube with v(S) > 0 and let f, g : S → R be two bounded functions.
(a)
Prove that for any two partitions P, P' of S, we have
(f + g) < U(f,P) + U(g, P'), and L(f,P)+ L(g,P') <
(f + g).
(b)
Prove that
f,g are integrable on S, then so is f + g, and
(ƒ + g)
f +
g.
(First use part (a) to prove that Js(f +g) < Tsf +J s9 and S(f+9) > Lgf + S,9-)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F09caa562-c0c8-4a11-85df-abb5e549e1a9%2F1bef9aa8-49c3-4bf0-9791-b4ff1ea39afb%2Fjl12fr_processed.png&w=3840&q=75)
Transcribed Image Text:#2 Let S be a closed n-cube with v(S) > 0 and let f, g : S → R be two bounded functions.
(a)
Prove that for any two partitions P, P' of S, we have
(f + g) < U(f,P) + U(g, P'), and L(f,P)+ L(g,P') <
(f + g).
(b)
Prove that
f,g are integrable on S, then so is f + g, and
(ƒ + g)
f +
g.
(First use part (a) to prove that Js(f +g) < Tsf +J s9 and S(f+9) > Lgf + S,9-)
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