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:

, C the arc of parabola which connect the point y

I've made a parametrization for C, thats easy:

And then I've made this integral:

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

But now I don't know if I should use the module, I did this: and I dont know when I should use this:

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.