According to Lipschutz's Differential Geometry book,
A surface of revolution $S$ is obtained by revolving a plane curve $C$ (the profile curve) about a line $L$ (the axis of $S$) in its plane. If $x_{1}=f(t)$, $x_{3}=g(t)$, $a<t<b$ is a regular curve $C$ of class $C^{m}$ in the $x_{1} x_{3}$ plane and $f>0$, then $\mathbf{x}=(f(t) \cos \theta) \mathbf{e}_{1}+(f(t) \sin \theta) \mathbf{e}_{2}+g(t) \mathbf{e}_{3}$, with $-\infty<\theta<\infty$, is a regular parametric representation of class $C^{m}$ of the surface obtained by revolving $C$ about the $x_{3}$ axis.
So the limits for the parameters would be $t \in (a,b)$ and $\theta \in (-\infty,\infty)$. However, as $\theta$ seems to be the azimuthal angle of the points in the surface, wouldn't indeed $\theta \in (0,2\pi)$ at most, rather than $\theta \in (-\infty,\infty)$?
In fact, for some cases such as the sphere, $\left(\cos \left(t\right)\cdot \cos \left(\theta\right),\cos \left(t\right)\cdot \sin \left(\theta\right),\sin \left(t\right)\right)$, it would be enought for $\theta$ to take the values in $(0,\pi)$.