5.8.7: Solid Cylinder
( \newcommand{\kernel}{\mathrm{null}\,}\)
Refer to figure V.8. The potential from the elemental disc is
dψ=−2πGρδz[(z2+a2)1/2−z]
and therefore the potential from the entire cylinder is
ψ=const.−2πGρ[∫h+lh(z2+a2)1/2dz−∫h+1hzdz].
I leave it to the reader to carry out this integration and obtain a final expression. One way to deal with the first integral might be to try z=atanθ. This may lead to ∫sec3θdθ. From there, you could try something like ∫sec3θ=∫secθdtanθ=secθtanθ−∫tanθdsecθ=secθtanθ−∫secθtan2θdθ=secθtanθ−∫sec3θ+∫secθdθ, and so on.