Flux of a Paraboloid without Parametrization

[waters]

Homework Statement

Find the outward flux of F = <x + z, y + z, xy> through the surface of the paraboloid z = x^2 + y^2, 0 ≤ z ≤ 4, including its top disk.

Homework Equations

double integral (-P(∂f/∂x) - Q(∂f/∂y) + R)dA
where the vector F(x,y) = <P, Q, R>
and where z = f(x,y) <-- f(x,y) is the function that undergoes partial differentiaion

The Attempt at a Solution

I didn't parametrize. My teacher wanted this method, where sum the surface integrals of two instances: a) when z = x^2 + y^2, and b) z = 4. My final answer was -32pi/3, with the flux at z = 4 being 0. I have a strong feeling I did something wrong but I don't know what. Please help. Any input will be greatly appreciated.

 

[LCKurtz]

Homework Statement

Find the outward flux of F = <x + z, y + z, xy> through the surface of the paraboloid z = x^2 + y^2, 0 ≤ z ≤ 4, including its top disk.

Homework Equations

double integral (-P(∂f/∂x) - Q(∂f/∂y) + R)dA
where the vector F(x,y) = <P, Q, R>
and where z = f(x,y) <-- f(x,y) is the function that undergoes partial differentiaion

The Attempt at a Solution

I didn't parametrize. My teacher wanted this method, where sum the surface integrals of two instances: a) when z = x^2 + y^2, and b) z = 4. My final answer was -32pi/3, with the flux at z = 4 being 0. I have a strong feeling I did something wrong but I don't know what. Please help. Any input will be greatly appreciated.

Why would you say the flux is 0 when z = 4? But in any case, if you have something wrong how are we supposed to help you when you didn't show us your work?