 Conical frustum

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.