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Math Help - Integral (continious decreasing functions)

  1. #1
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    Integral (continious decreasing functions)

    Hi im kinda stuck on this question, any hints are appreciated

    I got f:[1,infinity[ -> R, is a positive continious decresing function, and m < n for m,n belonging to N.
    I need to show that ∫ from m+1 to n+1 f(x)dx <= sum from k=1 to n of f(k) - sum from k=1 to m of f(k) <= ∫ from m to n f(x)dx

    Apologize for the setup, but hopefully its 'readable'
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  2. #2
    MHF Contributor matheagle's Avatar
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    Draw it and see the area of the regions.
    The widths are one, but since the function is decreasing
    the sum is between those two areas.

     \int_{m+1}^{n+1}f(x)dx \le \sum_{k=m+1}^n f(k)\le\int_{m}^{n}f(x)dx
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