
Line integral question
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 anticlockwise when viewed from the positive xaxis.
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.

With the parametric equations you give, t goes from 0 to (around the full circle), not 1 to 1.

Ahh, I see. So it wouldn't be to ? Or it doesn't matter?

Hmm, now I'm getting , which is also wrong. :/

I can't see where I'm going wrong, is the rest of the working correct?

the limits of your integration should be from to
(you are only tracing out a semicircle in the yzplane if C goes from (0,0,1) to (0,0,1) counterclockwise along the circle).

Thank you. I was a bit confused when HallsofIvy said 0 to 2pi.

Sorry, I misread the problem, thinking you were integrating around the entire circle. I wondered where you got 1 and 1 from!