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Riemann integral of arcsinh (have the answer, want an explanation)

  1. Jul 6, 2009 #1
    1. The problem statement, all variables and given/known data

    Given the following sum, turn it into an integral:
    [tex]\lim_{n \to \infty}\Sigma^n_{k=1}\dfrac{1}{n\sqrt{1+(k/n)^2}}[/tex]


    2. Relevant equations

    The answer says [tex]=\int^2_1\dfrac{1}{\sqrt{1+x^2}}[/tex]

    3. The attempt at a solution

    I understand how to get the equation, but why integrate from 1 to 2 and not from 0 to 1. if 1/n is the base length then the height should go from [tex]=\dfrac{1}{\sqrt{1+0}}[/tex] to [tex]=\dfrac{1}{\sqrt{1+1}}[/tex] not from [tex]=\dfrac{1}{\sqrt{1+1}}[/tex] to [tex]=\dfrac{1}{\sqrt{1+4}}[/tex]... or so i though??

    Thanks
     
  2. jcsd
  3. Jul 6, 2009 #2

    HallsofIvy

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    No reason I can think of! Clearly to identify [itex]1/\sqrt{1+ (k/n)^2}[/itex] with [itex]1/\sqrt{1+ x^2}[/itex] you have to take x= k/n. But with x= k/n, when k= 1 you have x= 1/n, which goes to 0 as n goes to infinity, and when k= n you have x= 1. The integral is from 0 to 1. Your book must have a typo.
     
  4. Jul 6, 2009 #3
    Ok cool thanks :)
     
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