Divergence theorem how do you calculate the volume integral, flow?

Question

Hello I really need a thorough answer to one task which is the following one:

Consider the vector field $$F(x, y, z) = (xy^2 z, x^2yz, −z),$$ the cylindrical-shape Area $$V = \left\{(x, y, z): \sqrt{x^2+y^2} < 5 \text{ and } 0 < z < 4\right\} $$

Now calculate the volume integral ,and calculate:

1. The flow through the surface of the cylinder,
2. The flow through the cover of the cylinder (the area at $z = 4$),
3. The flow through the bottom of the cylinder (the area at $z = 0$)

Sorry for not being accustomed to the formatting here,Im new here since today and I really need this task within the next 11 hours, Im stuck at it and Im not doing well regarding health. for reference it is task A37 on this link (sorry, it is german.)

I'd be very thankful if you could solve this.

Answer 1

Maybe this will work. Here is my answer