Double Integrals, polar coordinates, Stoke's theorem, and Flow line Questions

See the attached file and leave a comment if you need any clarifications. 

1- You would like to find the region $D$ on $xy$- plane which is the projection of the surface onto $xy$-plane. The region $D$ is bounded by $x=0$ and $x=1$. In order to find the bounds in $y$, you set $z=0$ to find the intersection of the surface $z=9-y^2$ with the $xy$-plane. Hence 
\[0\leq x\leq 1,  0\leq y\leq 3, 0\leq z \leq 9-y^2 .\]
 (see the graph). Then we just do a triple integral over this region. 

2- The double integral is over the shaded region D in the graph. 

3- We are using the formula for Surface Integrals, see the formula after the blue box in the following link: 
https://tutorial.math.lamar.edu/classes/calciii/surfintvectorfield.aspx

Problems 4, 5, and 6 are straightforward.