Arc length explanation

Question

I’m trying to understand the idea behind the arc length calculation. I understad the explanation connecting a finite amount of points and make a sum of the norm of all of them. But I don’t get why we can use the derivative of the point as the distance of the point itself. I get the main idea of it but as far as I know, the derivative indicates a direction not a longitude, so why are we using it for calculating a longitude?

Answer 1

Assuming that by "longitude" you mean "length": It isn't being used as a length. It's being used as a slope.

Roughly speaking, you slice your curve into a bunch of little pieces, and each piece is treated as the hypotenuse of a little right triangle, with horizontal leg $dx$ and vertical leg $dy$. So the hypotenuse has length $\sqrt{dx^2+dy^2}$. Then for the entire curve $S$, we have

$$ \operatorname{length}(S) = \int_S \sqrt{dx^2+dy^2}\,dS = \int_{x=a}^b \sqrt{1+\left(\frac{dy}{dx}\right)^2}\,dx $$