Compute the flux of F⃗ =3(x+z)i⃗ +4j⃗ +3zk⃗ through the surface S given by y=x2+z2, with 0≤y≤4, x≥0, z≥0, oriented toward the xz-plane.?

∫∫s F · dS
= ∫∫ <3x+3z, 4, 3z> · <y_x, 1, y_z> dA, with the normal pointing toward y=0
= ∫∫ <3x+3z, 4, 3z> · <2x, 1, 2z> dA, since y = x^2 + z^2
= ∫∫ (6x^2 + 6xz + 4 + 6z^2) dA.

Next, convert to polar coordinates (in x and z):
∫(r = 0 to 2) ∫(θ = 0 to 2π) (6r^2 + 6r^2 cos θ sin θ + 4) * (r dθ dr)
= ∫(r = 0 to 2) 2π (6r^2 + 0 + 4) * (r dr)
= ∫(r = 0 to 2) π (12r^3 + 8r) dr
= π(3r^4 + 4r^2) {for r = 0 to 2}
= 64π.

I hope this helps!