= Solution
The surface is obtained by revolving the graph $z=f(x)$ about the $x$-axis. It has a central cylindrical section of radius $b$, two smooth transition collars, and unit spherical caps centred at $(\pm3,0,0)$; its projection onto the $(x,z)$-plane is the region $|z|\leq f(x)$.
For a surface of revolution, the <Gaussian curvature of a surface of revolution> is
$$
K=-\frac{f''}{f(1+f'^2)^2}.
$$
\b[Thus the cylindrical region has $K=0$. On each transition collar, $K<0$ where $f''>0$, vanishes at the unique inflection circle, and is positive where $f''<0$. The spherical caps have $K>0$.]
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