Arc Length for Function Graphs
Text We can approximate the arc length of the function curve for a function f(x) by breaking the curve into segments of tangent lines.
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 √(dx^{2}+(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 = ∫_{a}^{b}√(1 + (f '(x))^{2})dx
Demos
Arc Length
 
Here is a visualization of approximation of arc length by segments of tangent lines. Increase the resolution to get a better approximation. Try changing f(x) as well.

Exercises 1. Use the demo to relate (using <, >, =) the arc lengths from 1 to 1 of the functions f(x) = 0, f(x) = x, f(x) = x^{2}, and f(x) = x^{3}. Explain from the formula for arc length why the comparison turns out this way.
2. How would one go about using the formula for arc length to find the arc length of a function which is not continuously differentiable (e.g. f(x) = x)?
