Integral through a path in 2D (or ND) What's the usual definition ?

[Swimmingly!]

Integral through a path in 2D (or ND) What's the usual "definition"?

[Bold letters are vectors. eg: r]
We have a scalar function f(r) and a path g(x)=y.
I see two ways to reason:
(1) The little infinitesimals are summed with the change of x and on the change of y separately.
(2) The little infinitesimals are summed with the change of r.

For example:
The scalar function is f(r)=1
The path is the straight line x=y, from x=0 to x=1.
(1) ∫dx+∫dy=1+1=2 ∫dx from 0 to 1, and since x=y, ∫dy from 0 to 1.
(2) ∫dr=√2 It's a straight path so ∫dr from 0 to √2.

What is the regular way to take an integral through a path?
(1) treats x and y totally independently, (2) seems more "physical/relative" but harder to calculate

 

[mfb]

A path integral is the integral along the length of the path, so (2).
I think (1) is not possible (in a meaningful way) for general paths.

 

[Vorde]

Like mfb said, 99% of the time when you're asked for a line integral of a scalar field you'll want it with respect to arc length, and then you'll want the integral with ds in it. As you showed in your post, $\int_C{F(x,y) dx}$ + $\int_C{F(x,y) dy}$ ≠ $\int_C{F(x,y) ds}$, so the line integral is defined like (2) in your post.

When you are doing line integrals in a vector field $\vec{F}(x,y) = <P,Q>$however, you'll find out that $\int_C{\vec{F}(x,y) \cdot d\vec{r}} = \int_C{P dx} + \int_C{Q dy}$, so then you'll use line integrals with regards to dx and dy.

 

[Swimmingly!]

Thank you for your answers.
I think it completely clears it up. (feel free to add anything if you want of course)