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Integration-Lower/Upper sum, Partition problem

  1. Jan 27, 2009 #1
    1. The problem statement, all variables and given/known data
    Suppose f is increasing and continuous on [a,b]. Show for any partition P,

    [tex] \int^b_a f(x)dx - L_f(P) \leq [f(b) - f(a)] \Delta x[/tex]

    2. Relevant equations
    Not sure if there are any . but for people unfamiliar with this notation:
    [tex] L_f(P) [/tex] = Lower sum for f with the partition P
    [tex] U_f(P) [/tex] = Upper sum for f with the partition P

    3. The attempt at a solution
    If you defined [tex] P = \{x_0, x_1, x_2,....,x_n_-_1, x_n\} [/tex] [tex]a = x_0[/tex] and [tex]b = x_n[/tex]

    I also know that
    [tex] L_f(P) \leq \int^b_a f(x)dx \leq U_f(P) [/tex]
    If i subtract The lower sum from both sides i get,

    [tex] 0 \leq \int^b_a f(x)dx - L_f(P) \leq U_f(P) - L_f(P)[/tex]

    which gives me the left side of what i'm trying to show. Now the problem arises trying to make the right side look like what it should. When the partitions are of equal size, the lower and upper sum's become a telescoping sum (I know this from a previous question), but in the given case, where the partitions can be of any size i can't find a way to make it look like the right side of the inequality. I've expanded both the lower and upper sums but was unable to cancel anything or find a way to manipulate it to make it look like [tex] [f(b) - f(a)] \Delta x[/tex].

    Any help would be great.
    Last edited: Jan 27, 2009
  2. jcsd
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