This is my first question in this site!. I create this question because I have a problem understanding how to use stokes's theorem in the next problem:
I need to verify the stokes's theorem in this vector field:
$$ F(x,y,z) = (z,x,y) $$
in the next surface: $$ Z = {x^2 + y^2 }~~under~the~plane~~Z= {2x} $$ I know that I need to prove: $$ \int_{\gamma}^{} (F\cdot T) dr = \int_S\int_{}(curl(F)\cdot n)~dS $$
In the first side for the curve I used: $$ \gamma(t) = \gamma_1(t) + \gamma_2(t) $$ where $$ \gamma_{1}(t) = (t,-\sqrt{2t-t^2},2t)~when~0 \leq t \leq 2 $$ $$ \gamma_{2}(t) = (-t,\sqrt{2t+t^2},-2t)~when~-2\leq t \leq 0 $$ with his derivative $$ \gamma_1'(t) = ({1,\frac{-1+t}{\sqrt{2t-t^2}},2}) $$ $$ \gamma_2'(t) = ({-1,\frac{1+t}{\sqrt{2t+t^2}},-2}) $$
and then I just did the line integral: $$ \int_{0}^{2} F(\gamma_{1}(t)) \cdot \gamma_{1}'~dt + \int_{-2}^{0} F(\gamma_{2}(t)) \cdot \gamma_{2}'~dt $$
But I don't like it this way to solve it, I think maybe some is wrong and is a little hard to do this integral (I'm pretty sure that there is a way to parameterize the curve easier with the intention to appear an easier integral).
In the other site, I have: $$ \int_S\int_{}(curl(F)\cdot n)~dS $$ where: $$ curl(F) = (1,1,1)~~and~n = (2x,2y,1) $$
But I don't know how to recognize my region S, so I can't continue in this path.
Any help is very helpful for me. Thanks to take your time in read my question!.