Integral Question (Riemann Sums)

[altcmdesc]

This is from a final exam on the MIT Open Course Ware site for Single Variable Calculus

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.

 

[Dick]

Divide the numerator and denominator by n^2. I.e. try to write the expression using only k/n.

 

[altcmdesc]

Ah, I get it. x/(x²-1). Thanks

 

[Dick]

Ah, I get it. x/(x²-1). Thanks

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