Calculating flux integral? Please help I need you!?

I would use the divergence theorem. The flux through a closed surface is found by integrating the divergence of the vector field, over the volume enclosed by the surface.

In your problem, the surface isn't closed, but that's OK, we can use the xy plane for the lower boundary, and consider separately any flux through that face. It will be much simpler than the flux through the hemispherical surface.

The divergence of any vector field is

du/dx + dv/dy + dw/dz,

where u,v, and w are the components of the field vector.

In your case the divergence is z + 0 + 0 = z.

So the outward flux across the closed hemisphere will be the volume integral of the function "z" over the solid hemisphere. The net flux across the base disk in the xy plane will be zero, because only the "k" components will cross this disk, and for every point with a positive "y" there is a corresponding point with a negative "y" of the same magnitude. Hence, the net flux across the upper surface will be the volume integral of z.

But this is easy. Slice the hemisphere into disks of thickness dz. The area of each disk is pi*(9-z^2). To integrate z dV, we just need to integrate from z = 0 to 3 the quantity

pi z (9-z^2) dz

= pi (9z - z^3) dz.

After integration you have

pi (9z^2/2 - (1/4)z^4), to be evaluated at z=3:

pi (81/2 - 81/4) = pi (81/4)....about 63.6

And that's the answer!