I need to calculate the **absolute value ** of the integral: $$ \oint_C (4z+2xy)dx + (x^2+z^2)dy+(2yz+x)dz $$ where $C$ is the intersection of the surfaces: $z=\sqrt{x^2+y^2 }, x^2+y^2 = 2y$ .
Will someone please help me find a suitable parameterization for this intersection curve? I have no idea how to do it.
Thanks !
We have that $x^2 + y^2 = 2y$ is equivalent to $x^2 + (y-1)^2 = 1 $, so $x = \cos t$ and $y = 1 + \sin t$ are good to go. From $z = \sqrt{x^2 + y^2}$ we get $z=\sqrt{2 + 2\sin t}$. So you can use: $${\bf r}(t) = (\cos t, 1+\sin t, \sqrt{2+2\sin t}),\quad 0\leq t \leq 2\pi.$$