I think I have returned all my math back to teachers without any refund. y=f(x); h=xb-xa, which is very small. My Q is to calculate curve length rather than area numerically. But let me use area as example to show you what i want. to calculate area between xa to xb, we have 2 ways: 1) area=(f(xa)+f(xb))*h/2; (trapezoid?) 2) area=(f(xa)+4*f(xm)+f(xb))*h/6; here xm=(xa+xb)/2; (parabola?) As my test, second one is much better than first. for curve length: 1) len=square root( (f(xb)-f(xa))*(f(xb)-f(xa)) + h*h); actually, it is distance from (xa, f(xa)) to (xb, f(xb)). do you know second way to calculate curve length as in area sample above, simple, easy-to-use and better? any links or explainations are highly appreciated. thanks. . .