If velocity vector is given as $\mathbf F=y\mathbf i +2 \mathbf j+\mathbf k$ , then find the flux of water through the parabolic cylinder $y=x^2$, $0\le x\le 3$, $0\le z \le 3$.
For this problem I think we need to construct a 3-dimensional graph on paper and then find the surface integral but I’m having an issue finding the unit normal vectors and in the whole integration process.
The surface can be parameterzied by x and z
$S = (x, x^2, z)\\ dS = \frac{dS}{dx} \times \frac{dS}{dz} = (2x,-1,0)\\ F = (x^2,2,1)\\ $
$\iint F\cdot dS = \int_0^3\int_0^3 2x^3-2 \, dx \, dz$
Think you can get home from there?