Hey, I just noticed that you referenced the book for the solution to problem 10.

That one's actually got a simple, if ass-backwards approach thanks to the
divergence theorem.

No word at this computer, sorry for the ascii.

Consider that
 _   _
dS = ndS

and
_              2   2   2
n = grad( f = x + y + z   ) = <2x,2y,2z>

And the unit normal is just 1/2 of that; 
                   _       _   _    2
So find a function F  s.t. F . n = x
         _
Simple!  F = 
                _
Notice that div F = 1

//  2     // _    _   ///     _      ///
|| x dS = || F . dS = ||| div F dV = ||| 1 dV
//        //          ///            ///
S         S            B              B

And that's just the volume of a unit sphere; 4pi/3!

I figured that was the point of the problem... perhaps the book should have
revisited that problem to show how much easier life can be with the divergence
theorem?