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(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
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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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-)
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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