Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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,

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

    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,

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

    Show all work.

    3. The attempt at a solution

    I've figured out a) to be:

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

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


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


    User Avatar
    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


    User Avatar
    Science Advisor
    Homework Helper

    Um, x/(x^2+1), right? You're welcome.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook