# Definite integral that is equal to Riemann sums

• May 5th 2012, 11:23 AM
Definite integral that is equal to Riemann sums
The problem is limit as n goes to infinity, riemann sum from i=1 to n of (2/n)(1+(2i/n))^1/2

The problem I have i getting rid of the ^1/2? I was thinking that I could evaluate the sums under the square root like Riemann sum of 1 + Riemann sum of 2i/n.all under the square root. I don't think that's how I'm suppose to do it.
Any help is greatly appreciated.

The 4 integrals to choose from are

integral from 1 to 2 f(x) = (1+x)^1/2dx
integral from 1 to n f(x) = (x)^1/2dx
integral from n to 2 f(x) = (1+x)^1/2dx
integral from 1 to 3 f(x) = (x)^1/2dx

What does the does the integral from 1 to the n mean?
• May 5th 2012, 11:53 AM
Plato
Re: Definite integral that is equal to Riemann sums
Quote:

The problem is limit as n goes to infinity, riemann sum from i=1 to n of (2/n)(1+(2i/n))^1/2

The problem I have i getting rid of the ^1/2? I was thinking that I could evaluate the sums under the square root like Riemann sum of 1 + Riemann sum of 2i/n.all under the square root.
Any help is greatly appreciated.
The 4 integrals to choose from are
integral from 1 to 2 f(x) = (1+x)^1/2dx
integral from 1 to n f(x) = (x)^1/2dx
integral from n to 2 f(x) = (1+x)^1/2dx
integral from 1 to 3 f(x) = (x)^1/2dx
What does the does the integral from 1 to the n mean?

I do not see in the list.
But $\int_1^3 {\sqrt {1 + x} dx}$ will work.

I think there is a typo in the last one.
• May 5th 2012, 11:55 AM
Re: Definite integral that is equal to Riemann sums
Thank you, but how did you figure it out.? I'm stuck with removing that square root.
• May 5th 2012, 12:10 PM
Plato
Re: Definite integral that is equal to Riemann sums
Quote:

Thank you, but how did you figure it out.? I'm stuck with removing that square root.

Look at this. But sure to click show steps.
• May 5th 2012, 01:08 PM
Re: Definite integral that is equal to Riemann sums
I know how to find the integrals but I don't know how to solve the Riemunns sums?
• May 5th 2012, 01:27 PM
Plato
Re: Definite integral that is equal to Riemann sums
Quote:

I'm stuck with removing that square root.

Quote:

The interval is $[1,3]$ its length is $3-1=2$. Divide $\frac{2}{n}$.
So the $\Delta_x=\frac{2}{n}$, and the partition points $x_k=(1+k\Delta_x)$ where $k=1,2,\cdots,n$
Thus $\sum\limits_{k = 1}^n {\sqrt {(1 + {x_k})} {\Delta _x}}$