Stokes Theorem, integrating a "cut" region

Question

I am having an issue with the two "Methods/Formulas" of Stoke's theorem. I am asked to evaluate a line integral. I have a curve C that is created by the intersection of the plane z=3-2x+y and cylinder x^2+y^2=4. For Method 1, I parameterized the curve and took an integral of F(r(t)) dot r'(t) dt to solve. No problem. I then tried Method 2 (to see what would happen), finding CurlF, finding partial derivatives for z=g(x,y), set up a polar double integral and attained the same answer as Method 1. The problem is that upon modelling this I see z=3-2x+y intersects the xy-plane interfering with my polar circle of radius 2, the area is cut by the plane (set z=0, the y=2x-3 line cuts the cylinder projection). My understanding is this cut negates the use of switching to polar and taking a double integral. I am getting the same answer, so am I just proving Stoke's Theorem? I've done tons of textbook questions but not one of them has a situation where the xy-plane is "cut" by the surface plane. So my work around is to always parameterize C and use Method 1 because I understand it better. I hope this is semi-clear:)

My concern is because of what Professor Leonard says in his lecture (Surface and Flux Integrals) https://www.youtube.com/watch?v=sQ0BJ3H-cZ8 at 3:23:00. The part about the are being dissected by the line x=2, and how that would affect the integration.

Thanks to MathLover for answering last time, I hadnt been around to update this and my post got removed.

Thanks

Answer 1

I also did both calculations and got the same answer of $28\pi$. Now here is why the Stoke's theorem curl integral makes sense. The line integral is integrating on the tilted elliptic boundary. The double integral is integrating on the interior of that tilted ellipse that is on the plane. The way we integrate on this plane is to parametrize the surface with polar coordinates. So instead of thinking the plane is cutting through the circle in the $xy$-plane with radius $2$, we should instead think of a map taking the circle of radius $2$ to the tilted ellipse on the plane because that is how we are able to integrate on surfaces (by parametrizing).

Hopefully that makes more sense, but do ask if you would like me to elaborate.