= Surface area of a surface of revolution
{title2=$2\pi\int\rho(t)\sqrt{\rho'(t)^2+z'(t)^2}\,dt$}
For the radial profile $(\rho(t),z(t))$, with $\rho>0$, use the <parametrized surface> $\mathbf R(t,\theta)=(\rho\cos\theta,\rho\sin\theta,z)$, $0\leq\theta<2\pi$. The two <tangent vectors> are <orthogonal> and have <Euclidean norms> $\sqrt{\rho'^2+z'^2}$ and $\rho$. Their <cross product> gives the displayed <surface area>. Geometric <surface area> assumes a single covering of the radial profile. A general three-dimensional generating curve must first be replaced by its actual <radius> $\rho(t)=\sqrt{x(t)^2+y(t)^2}$; its spatial <arc length> cannot simply replace the radial profile's <arc length>.
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