I want to evaluate ${\int}{\int_S}{\vec F}\cdot d{\vec S}$ with the given information in the title but cannot for the life of me figure this one out. I have looked at Stoke's theorem, and also the divergence theorem but they do not seem to fit, given my knowledge of multivariable-calculus.
I've been trying to use this fact: $${\int}{\int_S}{\vec F}\cdot d{\vec S}={\int}{\int_D}{\vec F} \cdot({\vec r}_u \times {\vec r}_v)dA$$ but I cannot figure out how to parameterize and also get the limits on integration for that, if it is even applicable. If I could figure out ${\vec r}(u,v)$ and the limits of integration then this problem is not hard but I can't.
If anyone could spare a few moments to help me I would really appreciate it. Sorry that I do not have any work to show but I am utterly stuck on this question and would love to solve it.
Thank you.
Here's an example sort of like mine but it is bounded top and bottom: Surface Integral over a cone
The surface $S$ is part of a rotational paraboloid. Use the following parametrization: $$S:\quad (u,v)\mapsto{\bf r}(u,v):=(u\cos v,u\sin v,u^2)$$ with parameter domain $D:=[1,\infty[\>\times\>[0,2\pi]$.
Make sure that ${\bf r}_u\times{\bf r}_v$ has $z$-component $>0$, and then compute the final integrand for your surface integral by plugging in the parametrization of $S$ into the definition of $\vec F$ and expressing ${\bf r}_u\times{\bf r}_v$ in terms of the parameters as well (that's what is called pullback): $$\psi(u,v):=\vec F\bigl({\bf r}(u,v)\bigr)\cdot\bigl({\bf r}_u(u,v)\times{\bf r}_v(u,v)\bigr)=u^3(u-4\cos v)\sin v\ .$$ The end result then would be $$\int_D\psi(u,v)\>{\rm d}(u,v)\ ;$$ but note that the last integral is only seemingly $=0$ by symmetry. In reality it does not exist since it is not absolutely convergent.