How to parameterize solid of revolution?

[Kuma]

Homework Statement

Here is the surface I need to parameterize. It is a solid of revolution.

Homework Equations

The Attempt at a Solution

So since its a piecewise function, I can define it as follows

(x-2)^2 + z^2 = 1, 1<x<2
z = -x+3, 2<x<3
z = x-3, 2<x<3

I know the formula for the area for a solid of revolution, but how do I parameterize this surface and then use that to calculate the surface area? I'm lost.

 

[LCKurtz]

I'll show you how to parameterize one piece. That should get you started. I would express the circular arc like this:$$
x=2+\cos\theta,\ z = \sin\theta,\ \frac \pi 2\le \theta\le\frac{3\pi} 2$$Now if you rotate that about the z axis, that will change the $x$ and $y$ values:$$
x=(2+\cos\theta)\cos\alpha,\ y=(2+\cos\theta)\sin\alpha,\ z=\sin\theta$$Now if you let $\alpha$ very from $0$ to $2\pi$, that will get the curved portion.

That's the general method. But if you don't have to do it that way, a probably simpler method would be to just rotate the ds arc elements around.