# Verifying Stokes' theorem with a line integral

I have the following question:

I've managed to do the surface integral part of stokes theorem (and got an answer of 0) however I'm having a lot of trouble doing the line integral part in order to verify the theorem.

As I understand it, I should be doing a line integral around the part of the plane $z = 2-2x-y$ that's in the $x,y,z > 0$ quadrant of the $x$ $y$ and $z$ axes but I don't know how to parametrize that particular part seeing as it's in 3D. All the examples I've done so far have been solely in the $xy$ plane.

Should I treat each side of the path I'm going to traverse as a separate line and then just do all four and add their answers up? Or am I missing something really simple?

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Since your boundary is where that plane intersects the coordinate planes, then you can just think in 2D by doing each part separately. You have the part in the xy-plane which is $2-2x-y=0$ by setting $z=0$. The part in the xz-plane by setting $y=0$ and getting the line $z=2-2x$. And lastly the part in the yz-plane by setting $x=0$ to get the line $z=2-y$.