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Math Help - Integral

  1. #1
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    Integral

    Hi, I have a question, but I am not sure about this.

    So we have the function $f(x)=1/x$ at the interval [1,2]. I have to find $0<\delta$ such that for every partition with $\lambda P<\delta$ such that
    $U(f,P)-L(f,P)<0.001$

    $\lambda P=max(x_k-x_k-1)$

    So what I did is to take partition with equal length, so that at the end of the calculation I would get something like 1/0.001<n and I could say something about the length $\lambda P$

    (I decided to upload a picture because it's hard writing here math)

    The solution that I wrote feels like a more specific case rather than general. I showed it for every partition with equal length although it is true for any partition. How can I solve this without having to chose partition with equal length?

    Thank you.
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  2. #2
    Super Member Rebesques's Avatar
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    Re: Integral

    For a random partition P, the series \sum_k f(x_{k-1})-f(x_k) is telescopic, so U(f,P)-L(f,P) equals a constant times \frac{1}{n}.
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