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Riemann Integral

  1. Mar 22, 2009 #1
    1. The problem statement, all variables and given/known data


    Suppose f(x):[a,b][tex]\rightarrow[/tex][tex]\Re[/tex] is bounded, non-negative and f(x)=0. Prove that [tex]\int^{b}_{a}[/tex]f=0.


    2. Relevant equations



    3. The attempt at a solution
    I am trying to use the idea that lower sums are zero, and show that the upper sums go to zero as the norm of the partition goes to zero. That is Upper sums [tex]\leq[/tex] c||P|| such that [tex]\int^{\overline{b}}_{a}[/tex]f=0.

    how can I prove that statement by using the idea above with the choice of c?
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Mar 22, 2009 #2

    matt grime

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    Are you saying that f is the zero function? I.e identically zero? (I only ask because you specify that f is bounded and non-negative, which would seem to be redundant.)

    Pick any partition. What are the Riemann sums for this partition? (This is trivial to work out, assuming I understand correctly that f(x)=0 for all x in [a,b]).
     
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