# Integral

• Jul 20th 2014, 10:29 AM
davidciprut
Integral

So we have the function $f(x)=1/x$ at the interval [1,2]. I have to find $0<\delta$ such that for every partition with $\lambda P<\delta$ such that
$U(f,P)-L(f,P)<0.001$

$\lambda P=max(x_k-x_k-1)$

So what I did is to take partition with equal length, so that at the end of the calculation I would get something like 1/0.001<n and I could say something about the length $\lambda P$

(I decided to upload a picture because it's hard writing here math)

The solution that I wrote feels like a more specific case rather than general. I showed it for every partition with equal length although it is true for any partition. How can I solve this without having to chose partition with equal length?

Thank you.
• Oct 9th 2014, 11:57 PM
Rebesques
Re: Integral
For a random partition $\displaystyle P$, the series $\displaystyle \sum_k f(x_{k-1})-f(x_k)$ is telescopic, so $\displaystyle U(f,P)-L(f,P)$ equals a constant times $\displaystyle \frac{1}{n}$.