Write the parametric equation of the surface generated by a parabola rotating around its axis.
I guess it's simply getting from the parabola equation to the parametric equations of a generic paraboloid. But I don't know how to get to the param. equations of that surface of revolution.
Any help would be really appreciated.
Like in other cases where the former $x$ axis becomes a rotating axis, it is natural to swith from $x$ to $r$ and to name the fixed vertical axis $z$ in replacement of $y$, i.e., transform $y=x^2$ into $y=r^2$ i.e., finally :
$$z=x^2+y^2\tag{1}$$
which is the cartesian equation of the paraboloid.
If you want parametric equations from (1), just take :
$$(x,y,z)=(x,y,x^2+y^2)\tag{2}$$
but if one prefers to take polar coordinates in the horizontal plane $x=r \cos \theta, y=r \sin \theta$, we obtain another parametric set of equations :
$$(x,y,z)=(r \cos \theta, r \sin \theta, r^2)\tag{3}$$