Evaluate the Flux with Divergence Theorem

[PsychonautQQ]

Homework Statement

Evaluate the flux where F = <(e^z^2,2y+sin(x^2z),4z+(x^2+9y^2)^(1/2)> in the boundary of the region x^2 + y^2 < z < 8-x^2-y^2

Homework Equations

The Attempt at a Solution

So using the divergence Theorem,

∇ dot F = 6

∫∫∫6r dzdrdθ

where z is bounded between r^2 and 8-r^2
and r is bounded between 0 and 8^(1/2)
and θ is bounded between 0 and 2∏

is this correct?
I'm mainly worried about my limits of integration for r and z are incorrect, can anyone verify?

 

[PsychonautQQ]

Okay so yeah what I wrote is probably wrong, can anyone help me and my friends along?

 

[PsychonautQQ]

BRAIN BLAST!
I just realized that the r bound should go from 0 to 2? right??

 

[CAF123]

BRAIN BLAST!
I just realized that the r bound should go from 0 to 2? right??

The enclosed volume is the volume between two paraboloids. I think what you have is correct.