I have a lot of problems with the way this is worded! First, the parameterization ofwhat? I presume you are talking about the parameterization of thesurfaceyou get by rotating the catenary.

You get the catenoid by rotating, not the parameterization!

But the original parameterization, (cosh(t), t) is in two dimensions. Which is the "x3-axis"? The only interpretation that makes sense (and gives the correct result) is that we really have (cosh(t), 0, t) in three dimensions.

Okay, rotating the curve around an axis gives a surface which requirestwoparameters. Since we already have "t", use that as one parameter and, since we are rotating, use the angle of rotation, , as the other. In polar (or cyindrical) coordinates, , . Since we are rotating around the x3 (z) axis, it is the first, x= cosh(t) that is our "r", or distance from the axis. Then and . Of course, z, measured along the axis of rotation, is not changed: z= t. That gives .