= Conical frustum
{title2=$\mathcal A_{\mathrm{lateral}}=\pi(a+c)\ell$}
The lateral surface obtained by rotating a straight segment in a <plane> containing the rotation axis has endpoint <radii> $a,c$ and slant length $\ell$. Its <surface area> is $\pi(a+c)\ell$: the <radius> is linear in the segment's <arc length>, so the <surface area of a surface of revolution> is $2\pi\int_0^\ell\rho(s)\,ds=2\pi\ell(a+c)/2$. This formula excludes the end disks and assumes that the segment's radial direction is fixed. A straight segment whose endpoint horizontal directions differ generally sweeps a different <surface of revolution>.
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