1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Riemann Sums - Help!

  1. Apr 11, 2012 #1
    1. The problem statement, all variables and given/known data
    Suppose f:[a,b] → ℜ is bounded and for each ε > 0, ∃ a partition P such that for any refinements Q1 and Q2 of P, regardless of how marked ⎟S(Q1,f) – S(Q2,f)⎟ < ε. Prove that f is integrable on [a,b].


    2. Relevant equations
    If P and Q1 and Q2 are partitions of [a, b], with P [itex]\subset[/itex] Q1, and P [itex]\subset[/itex] Q2, then Q1 and Q2 are refinements of P

    ∫(a to b) = sup {L(P,f)} so L(P,f) ≤ ∫(a to b)

    ∫(a to [itex]\overline{}b[/itex]) = inf{U(P,f)} so ∫(a to [itex]\overline{}b[/itex])≤ U(P,f)


    3. The attempt at a solution
    I know the S(Q1,f) and S(Q2,f) are squeezed between the Lower sum (P,f) and the Upper sum (P,f), and that the upper and lower Riemann integrals are less than or equal to the Upper and Lower sums. To prove f is integrable on [a,b] I have to show f is bounded, that was given, and show the upper and lower Rieman integrals are equal.
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Can you offer guidance or do you also need help?



Similar Discussions: Riemann Sums - Help!
Loading...