# 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:

Originally Posted by KenAdams
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 $\displaystyle \int_1^3 {\sqrt {1 + x} dx}$ 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:

Originally Posted by KenAdams
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:

Originally Posted by KenAdams
I'm stuck with removing that square root.

Quote:

Originally Posted by KenAdams
I know how to find the integrals but I don't know how to solve the Riemunns sums?

In your first reply you said that it was the square root giving you trouble.
Now you seem confused about Riemann Sums. In the future, please make your questions clear.

The interval is $\displaystyle [1,3]$ its length is $\displaystyle 3-1=2$. Divide $\displaystyle \frac{2}{n}$.

So the $\displaystyle \Delta_x=\frac{2}{n}$, and the partition points $\displaystyle x_k=(1+k\Delta_x)$ where $\displaystyle k=1,2,\cdots,n$

Thus $\displaystyle \sum\limits_{k = 1}^n {\sqrt {(1 + {x_k})} {\Delta _x}}$