This is from a final exam on the MIT Open Course Ware site for Single Variable Calculus(adsbygoogle = window.adsbygoogle || []).push({});

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.

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# Integral Question (Riemann Sums)

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