I cant really understand why it works this way. From adding up the lengths of small lines connecting two points on a curve to have the integral of scalar velocity from the beginning and end of the curve. https://snag.gy/4zPYc5.jpg (^ what i'm trying to say)
Can you find an intuitive notion to this?
EDIT: translation: The polygonal line obtained from the union of the pi's is an aproximation of the cirve c and its length is given by.... Considering smaller and smaller intervals you obtain a better aproximation.
Let alpha: [a,b] -> Rn be a regular curve of class c1. The arc length of the curve alpha from point t=a to t=b is....