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Homework Help: Proof with Darboux integral - question.

  1. Jan 14, 2013 #1
    1. The problem statement, all variables and given/known data
    I was asked to prove that for f(x), Riemann integrable and bounded from above by real number M:
    ∫[a,b] f(x) < M(b-a)

    2. Relevant equations

    3. The attempt at a solution
    Since f(x) is Riemann integrable, ∫[a,b] f(x) must be equal to both the lower and upper Darboux sums. Therefore: ∫[a,b] f(x) = supf(x) [a,b]Ʃ[i=1,n]Δxi = supf(x) [a,b](b-a) <= M(b-a)
    I am really not sure this is rigorous enough, or even how it ought to be approached. I'd appreciate some guidance please.
  2. jcsd
  3. Jan 14, 2013 #2
    I think this can be seen as a consequence of the integral mean value theorem.
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