Problem figuring out the surface of integration

[Amaelle]

Homework Statement

Consider the vector field F(x, y, z) = (2x, 2y, 2z) and the surface
Σ = n(x, y, z) ∈ R3 : y = 1 − 3x^2 − 3z^2 and x^2 + z^2 ≤ 9, x ≥ 0 ,
oriented so that its normal vector forms an acute angle with the fundamental versor of the y–axis.
Compute the flux of F through Σ.

Relevant Equations

K′ = [0, √3/3] × [-π2, π2 ]

Good day I have a problem figuring out the surface of integration
according to the exercice, we have a paraboloid that cross a disk on the xz plane, the parabloid cross the xz plane on a smaller disk r=√3/3

so for me after going to the final step of integration and using polar coordinate i will integer for a value of r going from 0 to √3/3
but the solution of the exercice said
If we parametrize K in polar coordinates, K is the image of K′ = [0, 3] × [-π2, π2 ]

Any help would be highly appreciated
thanks

 

[pasmith]

Nowhere does it say that $y$ is constrained to be positive; it says instead that $x^2 + z^2 \leq 9$ which is indeed $r \leq 3$.

 

[Amaelle]

Thanks a lot ! I got it now!