1. Line integral question

The question
For the vector field F$\displaystyle (x, y, z) = (x^2 + y^2 + z^2)$i - zj + (y + 1)k calculate $\displaystyle \int_C{F.dr}$ where the path C from (0, 0, -1) to (0, 0, 1) is the circle $\displaystyle y^2 + z^2 = 1$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)) = $\displaystyle (cos^2(t) + sin^2(t))$i - $\displaystyle sin(t)$j + $\displaystyle (cos(t) + 1)$k
= i - sin(t)j + (cos(t) + 1)k

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

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

= 2 + 2sin(1)

However the solution is $\displaystyle 2 + \pi$.

Where have I gone wrong? Thanks.

2. With the parametric equations you give, t goes from 0 to $\displaystyle 2\pi$ (around the full circle), not -1 to 1.

3. Ahh, I see. So it wouldn't be $\displaystyle -\frac{\pi}{2}$ to $\displaystyle \frac{3\pi}{2}$? Or it doesn't matter?

4. Hmm, now I'm getting $\displaystyle 2\pi$, which is also wrong. :/

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

6. the limits of your integration should be from $\displaystyle -\pi/2$ to $\displaystyle \pi/2$

(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).

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

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