real analysis – Flux through the positive part of a sphere centered at $(0,0,1)$

Let $$B={(x,y,z):x^2+y^2+(z-1)^2<4, zgeq 0}$$ and consider the vector field $$F:(x,y,z)mapsto(x^3,y^3,z)$$

I want to compute the flux of $F$ through $partial B.$

We have $$text{Div}F=3x^2+3y^2+1$$ and so by the divergence theorem we could compute the flux as $$int_B 3x^2+3y^2+1 dx dy dz$$
but this triple integral does not look very friendly to me using spherical coordinates, because the sphere is centered in $(0,0,1).$

Another approach is by computing $$partial B ={x^2+y^2leq 3, z=0}cup{x^2+y^2+(z-1)^2=4, z>0}$$

I can parametrize ${x^2+y^2+(z-1)^2=4, z>0}$ as $$x=2sin phi cos psi$$ $$y=2 sin phi sin psi$$ $$z= 2 cos phi +1$$ $$phiin(0,pi), psi in (0,2 pi)$$ but it does not seem to make calculations easier.

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Evaluation: 5

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How can i store =(-0.01 - 0.29)- (0 - 0) intact formula?

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