Line integral question

• June 2nd 2011, 06:55 AM
Glitch
Line integral question
The question
For the vector field F $(x, y, z) = (x^2 + y^2 + z^2)$i - zj + (y + 1)k calculate $\int_C{F.dr}$ where the path C from (0, 0, -1) to (0, 0, 1) is the circle $y^2 + z^2 = 1$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)) = $(cos^2(t) + sin^2(t))$i - $sin(t)$j + $(cos(t) + 1)$k
= i - sin(t)j + (cos(t) + 1)k

(1, -sin(t), cos(t) + 1).(0, -sin(t), cos(t))
= 1 + cos(t)

$\int_{-1 C}^{1}{1 + cos(t) \ dt}$

= 2 + 2sin(1)

However the solution is $2 + \pi$.

Where have I gone wrong? Thanks.
• June 2nd 2011, 06:58 AM
HallsofIvy
With the parametric equations you give, t goes from 0 to $2\pi$ (around the full circle), not -1 to 1.
• June 2nd 2011, 07:05 AM
Glitch
Ahh, I see. So it wouldn't be $-\frac{\pi}{2}$ to $\frac{3\pi}{2}$? Or it doesn't matter?
• June 2nd 2011, 07:22 AM
Glitch
Hmm, now I'm getting $2\pi$, which is also wrong. :/
• June 2nd 2011, 10:05 PM
Glitch
I can't see where I'm going wrong, is the rest of the working correct?
• June 2nd 2011, 11:25 PM
Deveno
the limits of your integration should be from $-\pi/2$ to $\pi/2$

(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).
• June 3rd 2011, 12:02 AM
Glitch
Thank you. I was a bit confused when HallsofIvy said 0 to 2pi.
• June 3rd 2011, 02:55 AM
HallsofIvy
Sorry, I misread the problem, thinking you were integrating around the entire circle. I wondered where you got -1 and 1 from!