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bodensee9

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**Please Help! Surface integrals**

I am wondering if someone can help me with the following? I am asked to evaluate ∫∫F∙dS where F(x,y,z) = z^2xi + (1/3y^3 +tanz)j + (x^2z+y^2)k and S is the top half of the sphere x^2+y^2+z^2 = 1.

∫∫F∙dS = ∫∫∫divFdV. Here, div F = x^2+y^2+z^2. I know that S is not a closed surface and so you would need to evaluate S as the difference between 2 surfaces, S1 as the closed surface that is the top half of the sphere and S2 as the disk that is x^2+y^2≤1 where the orientation is downwards. So, evaluating div F over the whole top half of the sphere I got 2pi/5.

But I am wondering how I would evaluate ∫∫∫x^2+y^2+z^2 over the disk x^2+y^2≤1? Could I convert to spherical coordinates and I could simplify the expression of the integrant to ρ^4sinφ, but I am wondering what φ would be in this instance? Thanks so much!