I'm working on a calculus 3 problem where I am asked to evaluate a line integral given initial and terminal points. However, the question specifies that I should let C be the upper half of the ellipse, where the intersection of the ellipsoid is $$ x^2+y^2+3z^2=3 $$ and the plane $$ y=\sqrt{2}x $$ What confuses me is how I should parametrize this ellipsoid. I'm assuming because they're asking for the top half of the ellipse, I should assume $$0\leq t \leq \pi$$ More specifically, does the direction matter when I'm parametrizing an ellipsoid (clockwise vs. counter-clockwise)? I'm also not that great at parametrization in general so I'm not sure if I'm on the right track or not.
What I have so far is
$$ x^2+y^2+3z^2=3$$ $$ y=\sqrt{2}x$$
which led me to
$$ x=\sqrt{3-y^2-3z^2}$$ $$ y=\sqrt{2}x$$
I'm unsure of how to proceed after this to complete my parametrization. Any help or tips would be greatly appreciated! Thanks!
Substitute $y^2 = 2 x^2$ into the equation of the ellipsoid and you get $3 x^2 + 3 z^2 = 3$, or $x^2 + z^2 = 1$, which is the usual equation of the unit circle in the $xz$ plane. One useful parametrization is $x = \cos(\theta)$, $z = \sin(\theta)$, and then $y = \sqrt{2} \cos(\theta)$.