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Integral Question (Riemann Sums)

  1. Aug 4, 2008 #1
    This is from a final exam on the MIT Open Course Ware site for Single Variable Calculus

    1. The problem statement, all variables and given/known data

    (a)(5 points) Write down the general formula for the Riemann sum approximating the Riemann integral,

    1
    [tex]\int f(x)dx[/tex]
    0

    for the partition of [0,1] into n subintervals of equal length. Evaluate the function at the right endpoints of the subintervals.

    (b)(5 points) Find a Riemann integrable function [tex]f(x)[/tex] on the interval [0, 1] such that the formula for the Riemann sum from (a) equals the following formula,

    n
    [tex]\sum \frac{k}{k^{2}+n^{2}}[/tex]
    k=1

    Show all work.


    3. The attempt at a solution

    I've figured out a) to be:

    n
    [tex]\sum \frac{f(k/n)}{n}[/tex]
    k=1

    Using this result, on b) I get as far as:

    [tex]f(k/n)[/tex]=[tex]\frac{kn}{k^{2}+n^{2}}[/tex]

    But I can't get any farther.
     
  2. jcsd
  3. Aug 4, 2008 #2

    Dick

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    Science Advisor
    Homework Helper

    Divide the numerator and denominator by n^2. I.e. try to write the expression using only k/n.
     
  4. Aug 4, 2008 #3
    Ah, I get it. x/(x2-1). Thanks
     
  5. Aug 4, 2008 #4

    Dick

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    Science Advisor
    Homework Helper

    Um, x/(x^2+1), right? You're welcome.
     
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