Need some help with a line integral problem...

Problem: Let C be the curve of intersection of the upper hemisphere $\displaystyle x^2 + y^2 + z^2 = 4, z\geq0$ and the cylinder $\displaystyle x^2 + y^2 = 2x$, oriented counterclockwise as viewed from high above the xy-plane. Evaluate $\displaystyle \int_{C}{ydx + zdy + xdz}$.

From what I understand, we need a parametrization in terms of one variable. I found an intersection curve parametrization as:

$\displaystyle x = 2 - \frac{t^2}{2}$

$\displaystyle y = \frac{t\sqrt{4 - t^2}}{2}$

$\displaystyle z = t$

And now I'm not sure how to proceed. I suspect I will need to evaluate this integral:

$\displaystyle \int_{t=a}^b {[\frac{t\sqrt{4 - t^2}}{2} * -t} + $ $\displaystyle t * (\frac{\sqrt{4 - t^2}}{2} - \frac{t^2}{2\sqrt{4 - t^2}})+ 2 - \frac{t^2}{2}] dt$

Assuming all of that is correct, I suppose my primary question is how to find the limits of integration.