Pretty straight forward, isn't it? If z= 4, your equation becomes so . That's a circle with radius 1.
The surface integral was 33pi. In part (b), I had to calculate volume integral where V is the region defined by and . This was 48pi.
Now I have to verify divegence theorem. Well, I need to show 33pi + contribution from z=4 + contribution from z=1 = 48pi.
Sorry if I was not clear enough.
In cylindrical coordinates, becomes . Of course, in order to have a closed figure, we must have the "top" and "bottom", z= 4, and z= 1.
The divergence theorem says that .
If you want to find the volume of the figure, , using that, you must have a vector function, , such that . What vector function did you use?
Of course, the top of the figure, z= 4, is the disk while the bottom of the figure, z= 1, is the disk which means that, to find the volume directly, you must integrate the constant 1 from z= 0 to z= 4 over the disk which is simply 1 times 4 times the area of that disk, . Then, integrate the constant 1 from z= 0 to (in cylindrical coordinates) for r= 1 to 2 (and from 0 to , of course).