I've been thinking about the mathematical motivation of the definition of line integrals and a crazy idea came to my mind.
First of all, let's begin with the definition of integral line of a function:
If $f$ is defined on a smooth curve $\gamma:[a,b]\to \mathbb R$, then the line integral of $f$ along $\gamma$ is
$$\int_{\gamma}f(x,y)ds=\int_a^bf(\gamma(t))|\gamma'(t)|dt$$
For me this definition is simply saying that the line integration is the one where the function is being integrate over a curve instead of a $x$-axis:
Note the $|\gamma'(t)|$ is the correction of the lengths of the partitions of the curves (the partitions are in the curve instead of the $x$-axis).
Am I right?