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. by Level 11 User (81.5k points)