line integral Parametrization Problem

Question

I am stuck with this question from my assignment based on line integral-

What I don't understand here is that given c(t), I can directly plug in values of x, y and z (respectively cost, sint and t/2π) and solve the integral for 0 < t < 2π but than what is the significance of joining (1 0 0) to (1 0 1) or is it like since line integral is independent of path joining the end points, answer in both cases will be equal?

Thanks in advance!

Answer 1

The grammatical subject of "joining" is the path, not you. That is, the problem isn't asking you to join those two points (e.g. by a straight line segment), it's merely describing the path as a path joining those two points. That's all there is to it.

Answer 2

You have to show the line integrals' curve is equal to zero, therefore it is irrotational. After that you know the vector $F$ is conservative and thus the line integral $F.dr= U(x_2,y_2,z_2)-U(x_1,y_1,z_1)$ which are the endpoints given to you.