The solution to this problem in the Study Guide For Vector Calculus (4th edition) is correct except for the following errors.
Part (a) is correct.
Part (b):
The integral that they set up for the volume is fine
but their integation is NOT fine. If you do the
integration correctly, you see that the volume
is 3pi R^4/2. This is always larger than
piR^4/2 though.
Part (c):
By Gauss's Theorem we see easily that the flux is -60piR^4k,
so we know their solution is wrong.
Thier mistake: on the side that makes contact with the
mountain,
(T_r cross T_t) dot k should be -r,
but they have r.
So the flux out this part should be
84 pi R^4k and not -84 pi R^4k.
So the total flux is then -60piR^4k.
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