Question

Find the flux of
F = x^3i  + y^3 j  + z^3 k  out of the closed surface
z = squroot (x^2 + y^2) a cone
0 ≤ z ≤ 4
oriented outward.

Answer 1

divergence theorem??

personal pref would be to set it up in cartesian as \(\iiint \, dx \, dy \, dz \quad div (\vec F) = \iiint dx\, dy \, dz \quad 3(x^2 + y^2 + z^2) \)

then switch to cylindrical: \(3\int_{0}^{2\pi}\, \int_{0}^{4}\, \int_{0}^{z}\, dr \, dz \, d\theta \quad (r^2 + z^2)r \)

or \(3\int_{0}^{2\pi}\, \int_{0}^{4}\, \int_{r}^{4}\, dz \, dr \, d\theta \quad (r^2 + z^2)r \).