What is the Flux through the surface?

[uchicago2012]

Homework Statement

Calculate the surface flux integral of F = 2xy xhat + (xz - 5xy2)zhat over the surface shown in the figure. Take n = k.
See Figure

Homework Equations

Flux = Integral (Electric Field * dA)

The Attempt at a Solution

So from the figure the total area is obviously a². The problem says the "surface flux integral" so I assumed they want the flux? For which you need to integrate what? Just integrating what they gave me seems wrong. I'm a bit confused about what was given and what is wanted. And I have no idea what n = k means, unless they mean the normal vector can be considered the z axis, which would make sense. But I'm still at a loss of what exactly I'm integrating.

 

[SammyS]

Homework Statement

Calculate the surface flux integral of F = 2xy xhat + (xz - 5xy²)zhat over the surface shown in the figure. Take n = k.
See Figure

Homework Equations

Flux = Integral (Electric Field * dA)

The Attempt at a Solution

So from the figure the total area is obviously a². The problem says the "surface flux integral" so I assumed they want the flux? For which you need to integrate what? Just integrating what they gave me seems wrong. I'm a bit confused about what was given and what is wanted. And I have no idea what n = k means, unless they mean the normal vector can be considered the z axis, which would make sense. Yes, that's what is meant! But I'm still at a loss of what exactly I'm integrating.

$$\vec{F}=2xy\,\hat{x}+(xz - 5xy^2)\hat{z}$$ Here is another unit vector in the z direction.

$$\int_{S}\vec{F}\cdot d\vec{A}=\int_0^a\int_0^a\ \left(2xy\,\hat{x}+(xz - 5xy^2)\hat{z}\right)\cdot\hat{z}\,dx\,dy$$