
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 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)) = $\displaystyle (cos^2(t) + sin^2(t))$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.

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

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

Hmm, now I'm getting $\displaystyle 2\pi$, 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 $\displaystyle \pi/2$ to $\displaystyle \pi/2$
(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!