Integral S Calculus 3 Question

Question

Evaluate ∬S​F⋅dS where F=−xi+2yj​−zk and S is the portion of y=3x2+3z2 that lies behind y=6 oriented in the positive y-axis direction.

Thank you in advance!

Answer 1

1) note y=g(x,z) so f(x,y,z)=y-g(x,z) where f is the surface

2) del f is the surface normal N

3) find the unit normal n and correct (if needed) for positive orientation (out of the region)

4) find Fdotn

5) change to polar

6) perform double integral Fdotn dA

7) I got 2π√2/3 but that is not verified