I did the following(for the first problem):

I imagined cylinder

,

First I consider the pipe without top and bottom (

and

)

Since

for

,

has components 1 and 1. The integral for pipe will be simply

Mr F says: I don't think so. See below.
Now I consider top and bottom discs: integral for bottom disc will be

because it's at

.The most interesting is top disc (

and

). For area of top disc I will have the following integral

but the top z is 1 so integral is

. dA is a little area at the top disc, and I think it is

where

So finally my integral is

Taking it from 0 to 1 I get

.

So the finall result is:

I have to submit both problems very soon. Can you check if I did first correctly? How should I prove second one?