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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.
     
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