I was studying for my multi-variable calculus test, when I realized that their seem to be two very similar ways to calculate the length.

(used to calculate arc length) vs (used to calculate length of normal vector/ any vector)

And I guess they really are basically the same, but I was wondering if someone could explain the difference, and perhaps why the first one is used in line integrals whereas you use the second in surface integrals.

Thanks!