Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Upper and lower Riemann sums

  1. Jun 23, 2010 #1
    Let [itex]f[/itex] be a Riemann integrable function defined on an interval [itex][a,b][/itex], and let [itex]P = \{a = x_0 < x_1 < \ldots < x_n = b\}[/itex] be a partition of [itex][a,b][/itex]. I don't understand why the definition of (say) the upper Riemann sum of [itex]f[/itex] associated with [itex]P[/itex] is always given as
    [tex]
    U(f,P) = \sum_{i=1}^n M_i (x_i - x_{i-1}),
    [/tex]

    where

    [tex]
    M_i = \sup\limits_{x\in [x_{i-1},x_i]} f(x).
    [/tex]

    Why does it need to be [itex]\sup[/itex]? Why can't it just be [itex]\max[/itex]? I can't think of an instance where [itex]\max[/itex] wouldn't be suitable; that is, I can't think of a Riemann integrable function that doesn't actually attain its max somewhere on any given [itex][x_{i-1},x_i][/itex]. Can someone provide an example for me?
     
  2. jcsd
  3. Jun 23, 2010 #2
    [itex]y=x[/itex] when [itex]x\neq 1[/itex], [itex]y=0[/itex] when [itex]x=1[/itex] on interval [itex][0,1][/itex]. Take interval [itex][x_{n-1},x_n][/itex] from the partition.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Upper and lower Riemann sums
  1. Lower and upper sums (Replies: 6)

  2. Upper and Lower Sums (Replies: 5)

Loading...