= Fixed-length stationary surface of revolution
{title2=$y(x)=k^{-1}(\cosh kx-\cosh ka),\quad l=2\sinh(ka)/k$}
The area of a <surface of revolution> about the horizontal axis is $2\pi\int|y|\sqrt{1+y'^2}\,dx$. For a curve below the axis with endpoints $(\pm a,0)$ and fixed <arc length>, its stationarity is equivalent to stationarity of $\int(y+\lambda)\sqrt{1+y'^2}\,dx$. The <Beltrami identity> gives $(y+\lambda)/\sqrt{1+y'^2}=C$. Choosing $C=1/k$, imposing the endpoints and integrating the length gives the displayed curve. The function $\sinh s/s$ increases strictly from $1$ to infinity on $s>0$, so every $l>2a$ determines exactly one positive $k$. Stationarity alone does not establish a global minimum.
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