Finding Arc Length of a Curve: Using ##dx## and ##dy##

[PFuser1232]

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 $dx$?

 

[mfb]

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.

 

[PFuser1232]

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.

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?

 

[mfb]

"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.