# Limit of Riemann sums

• Nov 16th 2012, 02:30 PM
nubshat
Limit of Riemann sums
Express the following integral as the limit of Riemann sums using the midpoint rule.
Do not evaluate it.

Attachment 25753

• Nov 16th 2012, 04:36 PM
Plato
Re: Limit of Riemann sums
Quote:

Originally Posted by nubshat
Express the following integral as the limit of Riemann sums using the midpoint rule
Attachment 25753

You are asking us to do this for you. NO WAY!
Note that $\displaystyle f(x)=\frac{x^2}{x^6+1}$ is an even function.
So the question reduces to $\displaystyle 2\int_0^1 {f(x)dx}$.

To partition the interval $\displaystyle [0,1]$ into $\displaystyle n$ regular subdivision we get $\displaystyle \Delta_x=\frac{1}{n}$.
Then $\displaystyle x_k=k\cdot\Delta_x,~k=0,~1,\cdots,~n$.
The midpoints are $\displaystyle m_k=\frac{x_k-x_{k-1}}{2},~k=1.\cdots,n$.
Now the midpoint sum is $\displaystyle \sum\limits_{k = 1}^n {f\left( {m_k } \right)\Delta _x }$
• Nov 16th 2012, 05:33 PM
nubshat
Re: Limit of Riemann sums
Thank you very much for the help
I got Attachment 25755 as an answer
Can someone please confirm that this is right or tell me what I did wrong?
edit: the x is suppose to be an n