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I am asked to find the surface integral ∫∫ydS where S is part of the paraboloid y = x^2+z^2 that lies inside the cylinder x^2+z^2 = 4.

The double integral could be rewritten as ∫∫y*√(4(x^2+z^2)+1)dS, or ∫∫(x^2+z^2)*√(4(x^2+z^2))dxdz. But this seems very difficult to integrate, so if I convert to polar coordinates, I should have ∫∫r^2*√(4r^2+1)rdrdθ, where r is between 0 and 2 and θ is between 0 and 2π. But I’m not really sure how to integrate r^3*√(4r^2+1)? Or did I set this up incorrectly? Thanks.