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This is from a final exam on the MIT Open Course Ware site for Single Variable Calculus
(a)(5 points) Write down the general formula for the Riemann sum approximating the Riemann integral,
1
\int f(x)dx
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 f(x) on the interval [0, 1] such that the formula for the Riemann sum from (a) equals the following formula,
n
\sum \frac{k}{k^{2}+n^{2}}
k=1
Show all work.
I've figured out a) to be:
n
\sum \frac{f(k/n)}{n}
k=1
Using this result, on b) I get as far as:
f(k/n)=\frac{kn}{k^{2}+n^{2}}
But I can't get any farther.
Homework Statement
(a)(5 points) Write down the general formula for the Riemann sum approximating the Riemann integral,
1
\int f(x)dx
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 f(x) on the interval [0, 1] such that the formula for the Riemann sum from (a) equals the following formula,
n
\sum \frac{k}{k^{2}+n^{2}}
k=1
Show all work.
The Attempt at a Solution
I've figured out a) to be:
n
\sum \frac{f(k/n)}{n}
k=1
Using this result, on b) I get as far as:
f(k/n)=\frac{kn}{k^{2}+n^{2}}
But I can't get any farther.