Surface element over sphere in Cartesian

Question

Suppose I want to evaluate a vector function f(x,y,z) over a hemisphere in Cartesian coordinates. $$\int \vec f.\vec{dS}$$ Over $x^2+y^2+z^2=1, z>0$. What would be my surface element. My research tells it should be something like $dS=dxdy \hat k + dydz \hat i + dxdz \hat j$. Then dot product would give me something like $f_x dydz + f_y dxdz+ f_z dxdy$. Then how would I set the limits. Is it correct?Pls guide me.