The question

For the vector fieldF$\displaystyle (x, y, z) = (x^2 + y^2 + z^2)$i- zj+ (y + 1)kcalculate $\displaystyle \int_C{F.dr}$ 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.