Line Integral of Vector Field without parametrization? In one of our problem sets, we were supposed to evaluate the line integral of the following vectorfield:
\begin{equation}\vec{F}(\vec{r}) = \sin(\pi x)\vec{e_x} -\cos(\pi y)\vec{e_y}\end{equation}
from points  $(1,\frac{1}{2}) \rightarrow (2,\frac{1}{2}) \rightarrow (2,1) $, $(1,\frac{1}{2}) \rightarrow (1,1)\rightarrow (2,1)$ and from (1,1/2) to (2,1) directly along the curve y=x/2 (all integrals are path independent, thats what we needed to show) 
The way it was solved was this:


*

*Path: \begin{equation}\int\limits_{(1,1/2)}^{(2,1)}\vec{F}(\vec{r}) = \int\limits_{1}^2 F_x(x)\,dx + \int\limits_{1/2}^2 F_y(y)\,dy\end{equation}


*Path: \begin{equation}\int\limits_{(1,1/2)}^{(2,1)}\vec{F}(\vec{r}) = \int\limits_{1/2}^1 F_y(y)\,dy + \int\limits_{1}^2 F_x(x)\,dx\end{equation}

*Path: \begin{equation}\int\limits_{(1,1/2)}^{(2,1)}\vec{F}(\vec{r}) = \int\limits_{1}^2 F_x(x)\,dx + \int\limits_{1/2}^2 F_y(y)\,dy\end{equation}
So in all three cases, it didn't seem like there was any parametrisation going on. So if I want to integrate a vector field along a curve, can I just integrate the x and y and z components seperately and add them all up?
 A: No.
In each of the above integrals, the two segments are being parameterized - by $x$ and $y$ respectively.  This works as long as you break up your path into segments along which your coordinates change monotonically.  
Explicitly, your first integral is 
$$\int_{(1,1/2)}^{(2,1)} F_x dx + F_y dy$$
The path has two segments.  The first is $(1,1/2)\rightarrow (2,1/2)$, parameterized by
$$ x = t, \ \ dx = dt$$
$$ y = 1/2, \ \ dy = 0$$
where $t\in [1, 2]$.  This yields the integral
$$\int_1^2 F_x(t) dt$$
The second segment is $(2,1/2)\rightarrow(2,1)$, parameterized by
$$ x = 2, \ \ dx = 0$$
$$ y = t, \ \ dy = dt$$
where $t\in [1/2, 1]$.  This yields the integral
$$\int_{1/2}^1 F_y(t) dt$$
For this choice of path, clearly it doesn't matter whether I call the integration variable $t$ or if I just re-use the coordinates $x$ and $y$, respectively.  I dislike re-using the coordinates for the most part, because doing so precludes us from using weird looping paths which cannot be straightforwardly parameterized by either coordinate.  I much prefer introducing the independent parameter $t$ which is generally applicable to whatever path you'd like.
