Line integral 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.
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 semi-circle in the yz-plane if C goes from (0,0,-1) to (0,0,1) counter-clockwise 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!