# Riemann integral of arcsinh (have the answer, want an explanation)

1. Jul 6, 2009

### foges

1. The problem statement, all variables and given/known data

Given the following sum, turn it into an integral:
$$\lim_{n \to \infty}\Sigma^n_{k=1}\dfrac{1}{n\sqrt{1+(k/n)^2}}$$

2. Relevant equations

The answer says $$=\int^2_1\dfrac{1}{\sqrt{1+x^2}}$$

3. The attempt at a solution

I understand how to get the equation, but why integrate from 1 to 2 and not from 0 to 1. if 1/n is the base length then the height should go from $$=\dfrac{1}{\sqrt{1+0}}$$ to $$=\dfrac{1}{\sqrt{1+1}}$$ not from $$=\dfrac{1}{\sqrt{1+1}}$$ to $$=\dfrac{1}{\sqrt{1+4}}$$... or so i though??

Thanks

2. Jul 6, 2009

### HallsofIvy

No reason I can think of! Clearly to identify $1/\sqrt{1+ (k/n)^2}$ with $1/\sqrt{1+ x^2}$ you have to take x= k/n. But with x= k/n, when k= 1 you have x= 1/n, which goes to 0 as n goes to infinity, and when k= n you have x= 1. The integral is from 0 to 1. Your book must have a typo.

3. Jul 6, 2009

### foges

Ok cool thanks :)