Let $\textbf{F} = (y+z,-xz,y^2)$. Let $S$ be the surface above the $xy$ plane and bounded by $2x + z = 6$, $y = 2$, $y = 0$ and $x = 0$. Calculate $$\iint_S \text{curl } \textbf{F} \cdot d\textbf{S}$$
I wanted to do this two different ways, one involving Stokes' theorem and the other just manually doing each face. However, I don't have any idea how to parameterise it for the manual case. When I tried using Stokes' theorem, I trace the boundaries $$(3,2,0) \to (0,2,6) \to (0,0,6) \to (3,0,0) \to (3,2,0)$$ counterclockwise, but that gives me $9 + 0 + 9 + 0 = 18$ which is wrong. Apparently the correct answer is $-6$. Any help would be greatly appreciated.
The answer $18$ should be the correct one. The four line integrals along the boundary of $S$ are correct: in the given order we find $9 + 0 + 9 + 0 = 18$.
The same result turns out by computing the flux of the curl: $$\begin{align}\iint_S \text{curl } \textbf{F} \cdot d\textbf{S}&= \iint_{[0,3]\times [0,2]} (2y+x,1,-(6-2x)-1) \cdot (2,0,1)\, dxdy\\ &=\iint_{[0,3]\times [0,2]}(4x+4y-7)\,dxdy\\ &=8[x^2/2]_0^3+12[y^2/2]_0^2-7\cdot 6=18.\end{align}$$
Moreover, in the statement, to be consistent with our calculations, the surface definition should be written in this way:
Let $S$ be the surface $2x + z = 6$ above the $xy$ plane and bounded by $y = 2$, $y = 0$ and $x = 0$. The orientation of $S$ is upward.
Then $S$ is a rectangle whose boundary is the union of four segments: $$(3,2,0) \to (0,2,6) \to (0,0,6) \to (3,0,0) \to (3,2,0).$$ It is oriented counterclockwise because $S$ is upward oriented.