5.8.7: Solid Cylinder

Refer to figure $\text{V.8}$. The potential from the elemental disc is

$$dψ = -2 \pi G ρ δ z \left[ \left( z^2 + a^2 \right)^{1/2} - z \right] \label{5.8.21} \tag{5.8.21}$$

and therefore the potential from the entire cylinder is

$$ψ = const. - 2 \pi G ρ \left[ \int_h^{h+l} \left(z^2 + a^2 \right)^{1/2} dz - \int_h^{h+1} z dz \right]. \label{5.8.22} \tag{5.8.22}$$

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 $z = a \tan θ$. This may lead to $\int \sec^3 θ dθ$. From there, you could try something like $\int \sec^3 θ = \int \sec θ d \tan θ = \sec θ \tan θ - \int \tan θ d \sec θ = \sec θ \tan θ - \int \sec θ \tan^2 θ d θ = \sec θ \tan θ - \int \sec^3 θ + \int \sec θ d θ$, and so on.