I have some trouble understanding every component of the line integral formula. Say I have a curve $c : [a,b] \mapsto \mathbb R^n$ and a scalar field $f : \mathbb{R^n} \mapsto \mathbb{R}$.
According to Wikipedia, the integral equation is then:
$$\int_c f \;ds = \int_a^b f(c(t)) |c'(t)| \;dt$$
I understand that $f(c(t))$ is the value of the scalar field on each point on the curve, and that $\int_c ds = \int_c |c'(t)|\;dt$ is the length of the curve.
Things I don't understand:
- What is $|h(x)|$, in general? Does it have any meaning outside the context of arc length?
- Is the result of the line integral the sum of all values of $f$ along the curve?...
- ... If yes, why is must we multiply $f$ by $ds$?