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?