Calculate $\int_{\Gamma} \omega$ when $\omega =z(z-y)dx+xzdy-xydz$
$\Gamma=\Gamma_1 \cup \Gamma_2 \cup \Gamma_3$
$$x^2+y^2=(z-1)^2$$
$x\geq0, y\geq0,z\geq0$
$\Gamma_{1,2,3}$ are the intersections with planes $x=0$, $y=0$, $z=0$
How do I find the right parametrization?
The paths over which the integral is taken look like this:
There is a circle at the bottom and four line segments protruding upwards. The parametrisation needs five smooth pieces, but that's fine – just add the results of the five integrals together.
The circle can be parametrised in the usual way as $(\cos t,\sin t,0)$ for $0\le t<2\pi$. The line segment from $(1,0,0)$ to $(0,0,1)$ can be parametrised as $$(1,0,0)(1-t)+(0,0,1)t=(1-t,0,t)\qquad0\le t<1$$ I leave it to you to find (very) similar equations for the remaining three line segments.