Curvilinear integral doubt

Hi. I have a doubt with this exercise. I'm not sure about what it asks me to do, when it asks me for the curvilinear integral. The exercise says:

Calculate the next curvilinear integral:

$\displaystyle \displaystyle\int_{C}^{}(x^2-2xy)dx+(y^2-2xy)dy$, C the arc of parabola $\displaystyle y=x^2$ which connect the point $\displaystyle (-2,4)$ y $\displaystyle (1,1)$

I've made a parametrization for C, thats easy: $\displaystyle \begin{Bmatrix} x=t \\y=t^2\end{matrix}$ $\displaystyle \begin{Bmatrix} x'(t)=1 \\y'(t)=2t\end{matrix}$

And then I've made this integral:

$\displaystyle \displaystyle\int_{-2}^{1}t^2-2t^3+(t^4-2t^3)2t dt$

But now I'm not too sure about this. What I did was:

$\displaystyle \displaystyle\int_{a}^{b}F(\sigma(t))\sigma'(t)dt$

But now I don't know if I should use the module, I did this: $\displaystyle \displaystyle\int_{a}^{b}F(\sigma(t))\sigma'(t)dt$ and I dont know when I should use this: $\displaystyle \displaystyle\int_{a}^{b}F(\sigma(t)) \cdot ||\sigma'(t)||dt$

I mean, both are curvilinear integrals, right?

I think that I understand what both cases means, but I don't know which one I should use when it asks me for the "curvilinear integral". The first case represents the area between the curve and the trajectory, and the second case represents the projection of a vector field over the trajectoriy, i.e. the work in a physical sense, but I know it have other interpretations and uses.

Well, thats all. Bye there, thanks for posting.