LAST EDITED ON Aug-20-01 AT 11:23 PM (EST)

>and possible solution would
>be to divide the graphic
>in small intervals, and take
>the distance from each of
>their borders or from the
>center of one to the
>next, and add up all
>this results, nd make the
>intervals every time smaller, but
>how do you express that
>in mathematic language?.
>

Your basic idea is right. This is what differential and integral calculus are about.

In short, the length of the graph of a function y = f(x) can be calculated as the integral of the interval of interest of the square root of the expresssion 1 + (f')2, f' is the derivative of the function f.

Where does such a formula come from?

On any small interval dx, as you suggested, consider the function change df and, according to the Pythagorean theorem, compute the hypotenuse - approximate length of the graph over dx:

sqrt(dx2 + df2) = sqrt(1 + (df/dx)2)dx.

If intervals dx cover the whole interval, say [a,b], with no interlapping, then the sum of the above expressions is close to the above mentioned integral. The expression df/dy, on the other hand, tends to the derivative as the intervals become smaller and smaller.