Find the flux of f = x^3 i + y^3 j + z^3k through the closed surface bounding the solid region x^2 + y^2 ≤ 4, 0 ≤ z ≤ 4

Use the divergence theorem. Let be the cylindrical region, then



(where denotes the unit normal vector to , but we don't need to worry about it now)

We have




For the solid with boundary , we can set up the following volume integral in cylindrical coordinates for ease:

Cis your correct answer