A cylinder is not always can-shaped? I am reading:

A right cylinder with cross section $\Omega$ is a solid that is formed by translating $\Omega$ along a line, or _axis_, that is perpendicular to it.

So my understanding now is a cylinder is any solid that can be formed by translating a 2D shape straight along an axis.
So extrusion forms a "cylinder", regardless of shape?  What do you call a "cylinder with a hole in it" (one whose cross sectional area shape has a hole in it)?
 A: The authors of that calculus book are giving you their definition of a cylinder for the purposes of working an integral. Certainly, an extrusion would be an effective word here, but not a word that would be widely recognized. An alternative term might be prism, but prism evokes the idea of $\Omega$ being a polygon. It is the Humpty Dumpty principle. To paraphrase: a word means what I say; nothing more, nothing less.
A: There is no definitive answer to what restrictions there must be; "cylinder" can mean what a given author so chooses.  Standard cylinders take $\Omega$ to be a circle (if referring to the surface) or a disk (if referring to the filled in "solid").  The Wikipedia article calls the resulting surface a "generalized cylinder" in the case where $\Omega$ is allowed to be an arbitrary curve, and an "oblique cylinder" if you relax the condition of translating perpendicularly.  It also states:

In differential geometry, a cylinder is defined more broadly as any ruled surface spanned by a one-parameter family of parallel lines. A cylinder whose cross section is an ellipse, parabola, or hyperbola is called an elliptic cylinder, parabolic cylinder, or hyperbolic cylinder respectively.

I guess I would call your solid cylinder with a hole in it an "annular cylinder", or simply the space between two cylinders with a common axis (if that is the shape you have in mind).
