$\int_C z dx+xdy$ where $C= \{(x,y,z): x^2+y^2+z^2=4, 2z+x=0\}$. I know how to calculate the integral but I'm not sure how to parametrize the set $C$.
Any help would be appreciated, I was thinking of using spherical coordinates but the parametrization of spherical coordinates would not be based on a single variable $t$.
Hint: You know that $x=-2z$. Plug this into the equation of the sphere to get the curve of intersection.
Answer: $x=-2z$ Then $(-2z)^2+y^2+z^2=4$ This is equal to $5z^2+y^2=4$ Leads to the parametrization of an ellipse so $z=sqrt(4/5)sin(t)$ and $y=2 cos(t)$ and $x=-2z=2sqrt(4/5)sin(t)$