# 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 $z=2-y$.

Parametrize those separately and do the integral. The last thing to be careful of is that when you add the integrals you may get a sign wrong if you don't parametrize according to the orientation. I'd maybe sketch it, so that the orientation is clear, and then just make sure the parametrization starts and ends in the right place.

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Thanks Matt, really appreciate it! Managed to get the right answer. I originally thought that the path would look a bit like a rectangle whereas I think the path you described looks like a triangle with vertices in the three coordinate planes? I guess that makes sense as if it were a rectangle then the corners in the zy and xz planes would have negative coordinates but I have a really hard time visualising these things in 3D. –  user6462 Feb 1 '11 at 3:49