Further solutions were submitted by Philippe Fondanaiche, Nancy Schwarzkopf, Bill Webb, Rick Bischoff, Lou Cairoli, Sudipta Das, Alejandro Vellano, Paul Botham.

Although purists will cringe, the following very simple "infinitesimal argument" by Nathan Pauli shows that the only function with the required property must be a scalar multiple of the simple quadratic.

The volume generated by an infinitesimal piece dx revolved about the y-axis is 2p x y dx.  The volume for the corresponding infinitesimal piece dy is p x² dy.  Equating and manipulating the terms gives the differential equation dy/y = 2 dx/x.  Integrating now gives ln(y) = 2 ln(x) + C, for an arbitrary constant C, which yields y = A x², for an arbitrary constant A.

The purists among us can quickly fix up this argument to make it "rigorous", though no more "correct".

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