1. Limit of Riemann sums

Express the following integral as the limit of Riemann sums using the midpoint rule.
Do not evaluate it.

2. Re: Limit of Riemann sums

Originally Posted by nubshat
Express the following integral as the limit of Riemann sums using the midpoint rule
You are asking us to do this for you. NO WAY!
Note that $f(x)=\frac{x^2}{x^6+1}$ is an even function.
So the question reduces to $2\int_0^1 {f(x)dx}$.

To partition the interval $[0,1]$ into $n$ regular subdivision we get $\Delta_x=\frac{1}{n}$.
Then $x_k=k\cdot\Delta_x,~k=0,~1,\cdots,~n$.
The midpoints are $m_k=\frac{x_k-x_{k-1}}{2},~k=1.\cdots,n$.
Now the midpoint sum is $\sum\limits_{k = 1}^n {f\left( {m_k } \right)\Delta _x }$

3. Re: Limit of Riemann sums

Thank you very much for the help