Originally Posted by

**yvonnehr** Can someone please help me prove this? I've tried it to work it out in different ways and I'm beat. Will appreciate your help.

The problem reads:

Let $\displaystyle f$ be a continuous function on $\displaystyle [a,b]$. The length $\displaystyle l$ of the curve $\displaystyle y=f(x)$ on the interval $\displaystyle [a,b]$ is defined by

$\displaystyle l=sup_{\substack{P}}[\lambda_P(f)]$

(don't know how to the _P under the sup)

where

$\displaystyle \lambda_P(f)=\sum_{k=0}^N\sqrt{(x_k-x_{k-1})^2+(f(x_k)-f(x_{k-1}))^2}$

and the supremum is taken over all partition points $\displaystyle D=\{a=x_0,x_1,...,x_N=b\}$ of $\displaystyle [a,b]$.

Assume that $\displaystyle f$ has continuous derivative on $\displaystyle [a,b]$. Take any partition $\displaystyle D=\{a=x_0, x_1,...,x_n=b\}$ of $\displaystyle [a,b]$ and derive

$\displaystyle \lambda_P(f) = \sum_{k=0}^N$ $\displaystyle \sqrt{1+\left(\frac{f(x_k)-f(x_k-1)}{x_k-x_{k-1}}\right)^2}$ $\displaystyle (x_k-x_{k-1})$.