Let's imagine the graphic of the sine function, and we take an interval from . if we take a string, or a metric tape, and measure the graphic, what would be its length?. what about other functions? if we take a linar function, it is easy, you just use the formula L(sq)=(a-b)sq + (f(a) - f(b))sq. but it does not work like that with other graphics. ane 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?.

thanks

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.