Gravitational attraction of solids of revolution: Part 1: Vertical circular cylinder with radial variation of density

Integral expressions are derived for the vertical component of gravitational attraction arising from vertical circular cylinders and horizontal circular disks with radial variation of density. The expressions contain a single elliptic integral of either the first or second kind and are evaluated easily by numerical integration. Various closed-form solutions are obtained for special cases. For constant density, the expressions reduce to well-known solutions as well as approximations that are based on the solid angle and line element, thus providing a unified theme for the derivations of those solutions and approximations. Numerical implementation is discussed in terms of practical application.