If $${\bf F}=2y{\bf i}-z{\bf j}+x^2{\bf k},$$ and $s$ is the surface of the parabolic cylinder $y^2=8x$ in the first octant, bounded by the planes $y=4$ and $z=6$, evaluate $$\int_S{\bf F}\cdot{\bf\hat n}\,dS,$$ where $\bf\hat n$ points in the direction of increasing $x$, by projecting the integral onto the plane $x=0$.
I'm trying to draw a sketch to get a feel of the situation but am confused as to what the question is asking. I have sketched $y^2=8x$ in the plane $z=0$ and marked on the points where $y$ and $z$ are bounded.
Here is a drawing of the situation . The surface $S$ is shown in tan, the yellow arrow points in the direction of increasing x . The unit normal to the surface is the vector shown in red , the plane $R$ , $x=0$ is shown in white. I believe you need to develop a projection formula of the form : $$ \int_S F \cdot \hat n dS = \int_R \frac{F \cdot \hat n}{\| \hat n \cdot \mathbf i \| } dR $$ where $ \mathbf i $ is the unit x vector . ( The general formula should be in your text , we are choosing the x rather than z coordinate vector, alternately you can permute indices and use the standard projection formula.)