A segment that starts at the point (x, f(x)) and ends a distance dx to the right will end at the point (x + dx, f(x + dx)) = (x + dx, f(x) + f '(x)dx). This means the length of the segment is √(dx2+(f '(x)dx)2). This expression simplifies to √(1 + (f '(x))2)dx.
Integrate with respect to x from a to b, resulting in the following formula for arc length s:
s = ∫ab√(1 + (f '(x))2)dx