The answer to your question really needs mathematical equations and formulae to express what is going on - as in a Word document using the mathematics add-in. I have answered your question in a Word document but can't upload it into the answer box, so I will have to try and give a verbal description.

The locus of a point on the curve C can be given by: r = xi + yj, where r is a vector position and x is a function of something or other and y is a function of something or other else.

Since we have r = ti + t^2j, then we can write x = t and y = t^2.

And, for later use, dr/dt = 1.i + 2t.j

We can also write, F•(dr/dt) = (cos(t).i +sin(t).j)•(1.i + 2t.j) = cos(t) + 2tsin(t)

Using F•dr = F•(dr/dt)dt

then

int{ F•dr) = int{F•(dr/dt)dt} =int{cos(t) + 2tsin(t)}[t=-1 to 2] dt

int{ F•dr) = {sin(t) - 2tcos(t) + 2sin(t)}[t=-1 to 2] = {3sin(t) - 2tcos(t)}[t=-1 to 2]

Ans = {3sin(2) - 4cos(2)} - {3sin(-1) - 2cos(-1)}

**Ans = 3(sin(2) + sin(1)) - 2cos(2) - 2(cos(2) - cos(1))**

The above expression could probably be simplified, using sum of sines and difference of cosines, but it probably wouldn’t be that much simpler, to read.