Using the Divergence Theorem, following the steps outlined below.

Let $S$ be the surface made up of the cylinder $x^2+y^2=4,0z10$ along with a top disk at height $z=10$ and a bottom disk at height $z=0.$ This surface looks like a tin can and includes the top and bottom pieces. You are asked to compute the flux for the vector field outward through $S.$

$($a$)$ Sketch $S.$ How many separate faces are there? Do not sketch the vector field.

Using surface integrals to find the total flux, we would need a separate flux integral for each face.

Instead of computing more than one flux integral, it is easier to do the problem using the divergence theorem. Goal: set up and compute $div(F)dV$ for our "tin can" shape, using the following steps:

$($b$)$ Compute the divergence, div$(F).$

$($c$)$ Find the limits of integration for the volume enclosed by the surface and set up the integral. Use any coordinate system of your choosing.

$($d$)$ Find the total flux by computing the integral and box your final answer.