Rotating a 3-dimensional curve $z=2+\sin y$ about the y-axis and parameterizing it. 
Consider the curve $z=2+\sin y$, $0 \leq y \leq 4\pi$ in the $yz$-plane. Find a parameterization for the curve rotated about the $y$-axis.

I understand that originally the distance to the y-axis is going to be our $z$ value. After the rotation about the y-axis the distance will be given by a point $(x,z)$ and so it will be of the form $\sqrt{x^2+z^2}$.
I substitute $z$ for $\sqrt{x^2+z^2}$ and get $\sqrt{x^2+z^2} = 2+\sin y$ but am unsure as to how to proceed with the parameterization.
 A: We can think of the curve as a function $f:R \to R^3$ given by $f(t) = (0, t, 2 +\sin t)^\top.$  The curve can be rotated by $\theta$ radians about the $y$ axis via the a rotation matrix.  In particular, if $g (t)$ is the rotated curve, then it is given by
$$
g(t) = \begin{bmatrix} \cos \theta & 0 & \sin \theta \\ 0 & 1 & 0 \\ -\sin \theta & 0 & \cos \theta \end{bmatrix} f(t) = ((2 +\sin t)\sin \theta, t, (2 +\sin t)\cos \theta)^\top
$$
Note that you could rotate the curve by $-\theta$ to get $ (-(2 +\sin t)\sin \theta, t, (2 +\sin t)\cos \theta)^\top.$
Edit:
Mathematica command for plot3D is here:
$\color{blue}{\text{ParametricPlot3D}[\{\sin (\theta ) (-(\sin (t)+2)),t,\cos (\theta ) (\sin (t)+2)\},\{t,0,4 \pi \},\{\theta ,0,2 \pi \},\text{Axes}\to \text{None},\text{PlotStyle}\to \text{Yellow}]}$

A: Hope everything is self-explanatory with Mathematica plot command. 
EDIT 1:
(Missed uploading requested plot!)
When rotated about $y$ axis $ x,z$ should be interchangeable (Fig at left).
$$ (z,x) = (2+ \sin \,y ) $$
ParametricPlot3D[ {  (2 + Sin[u]) Cos[t], 
  u, ( 2 + Sin[u]) Sin[t]}, {u, 0, 4 Pi}, {t, 0, 2 Pi}, Axes -> None, 
 Boxed -> False, PlotStyle -> Yellow]

When rotating about $z-$ axis $x,y$ should be interchangeable. (Fig at right). 
$$ z = 2 + \sin \sqrt{x^2+y^2} = 2 + \sin r $$

