With the parametric equations you give, t goes from 0 to (around the full circle), not -1 to 1.
The question
For the vector field F i - zj + (y + 1)k calculate where the path C from (0, 0, -1) to (0, 0, 1) is the circle in the plane x = 0, the direction of motion being anti-clockwise when viewed from the positive x-axis.
My attempt
Let z = sin(t), y = cos(t)
C(t) = (0, cos(t), sin(t))
C'(t) = (0, -sin(t), cos(t))
F(C(t)) = i - j + k
= i - sin(t)j + (cos(t) + 1)k
(1, -sin(t), cos(t) + 1).(0, -sin(t), cos(t))
= 1 + cos(t)
= 2 + 2sin(1)
However the solution is .
Where have I gone wrong? Thanks.