Use the Divergence Theorem to evaluate $ \iint_{S} \vec{F} \cdot \hat{n}\,dS, $ where $ \vec{F} = \langle 4x, 2y^2, z^2 \rangle, S $ is the boundary of the region defined by $ x^2 + y^2 + z^2 \leq 4,\,\, 0 \leq z \leq 3 $ and $\hat{n} $ is the unit outward normal
Attempt: The word 'boundary' is confusing me a little. So the surface is the curved part of a cylinder from $z=0$ to $z=3$. In cylindrical coordinates, $ x = 2\cos\theta, y = 2\sin\theta, z = z$. Using the Div. Thm gives $\text{div} \vec{F} = 4 + 4y + 2z $, so I have $ \iiint_{E} 4+4y+2z\,dV,$ Am I correct to write this as $ \int_{0}^{2\pi} \int_{0}^{2} \int_{0}^{3} (4 + 8\sin\theta + 2z)\,r\,dz\,dr\,d\theta? $ I am not sure because the question wanted the surface to be just the boundary but here I am integrating over the whole surface. Then again, I think this could be a feature of the div theorem. If we are just integrating on the boundary, then what would dV be? Many thanks