Evalute this integral using stoke's theorem

Question

$\int \bar F \cdot d \bar r $where $\bar F=(x-y)i+(y+z)j+xk $ and $C$ is boundary of the area of the triangle cut-off from the plane $2x+y+z=2$ by the coordinate axis. I tried to solve this question-Stoke's theorem $$ \int \bar F \cdot d\bar r=\iint \bar N \cdot (\nabla \times \bar F)ds$$ So i got $$\bar N=\frac {(2i+j+k)}{\sqrt 6}$$ and $$\nabla \times \bar F=-i-j+k$$ $$\iint \bar N \cdot (\nabla \times \bar F)ds=\frac {-2}{\sqrt 6}\iint ds $$ but i am stuck here what to do next....

Answer 1

Note that $$\iint_Sds = Area(S)$$ so you should just calculate the area of the triangle, which shouldn't be too hard.

Hence, the solution will be $$\frac{-2}{\sqrt{6}}Area(S)$$

with $S$ the area of the triangle cut-off from the plane by the coordinate-axis.