# Arc length

1. Mar 14, 2015

The arc length of any curve defined by $y = f(x)$ is found as follows:
$$ds = \sqrt{dx^2 + dy^2}$$
$$ds = \sqrt{dx^2(1 + {\frac{dy}{dx}}^2)}$$
$$ds = \sqrt{dx^2} \sqrt{1 + [f'(x)]^2}$$
$$ds = \sqrt{1 + [f'(x)]^2} dx$$
Isn't $\sqrt{dx^2}$ equal to $|dx|$, and not $dx$?

2. Mar 14, 2015

### Staff: Mentor

In principle it is, but if you go in positive x-direction only it does not matter. That way of dealing with differentials is not very mathematical anyway.

3. Mar 15, 2015

I'm not very familiar with the notion of "going in the positive direction" while plotting a function, I had no idea it makes a difference. Could you please elaborate?

4. Mar 15, 2015

### Staff: Mentor

"dx>0"

If you want to know the arc length between x=2 and x=4 for example, you integrate x from 2 to 4 and not from 4 to 2.