Flux of vector field through a surface.

I'm I'm calculating the flux of

v = (y, x)

upwards across the upper half of the unit sphere. (sphere such that z >= 0)

Is there two possible ways of going about this:

Find N = ( - (partial derivative function wrt x) , - (particial derivative wrt y, 1, )

the integrate v dot N with respect to x and y with the following limits:

OR

Parameterize S using spherical polar coordinates

Find N as the cross product between the partial derivative wrt theta and phi.

Then integrate v dot N with respect to theta and phi with the following limits:

I THINK this is correct but I'm a little confused because in the first case I integral with limts corresponding to a unit circle in R2 but in the second case it seems like my limits are drawing out the actual unit sphere?

Sorry for the lack of LateX I tried but failed to make it work!!

Anyhelp really appreciated!