$\oint_C 2xydx + [(1-y)z+x^2+x]dy + (\frac{x^2}{2} + e^z)dz$,
where $C$ is the intersection of the cylinder $x^2+y^2=1$, $z\ge0$ with cone $z^2=x^2+(y-1)^2$.
Although I managed to compute it sketching the graph, I don't understand why this happens when the algebraic path is taken for finding the intersection curve:
$x^2=1-y^2$
$z^2=x^2 + (y^2-2y+1)\rightarrow z^2=(1-y^2) + (y^2-2y+1)$
$z^2= 2 - 2y$
$y = -\frac{z^2}{2} + 1, z\ge0.$
This curve is supposed to be closed. What is the mistake here?