The discussion centers on proving Archimedes's principle for a sphere and a cylindrical vessel of the same volume, focusing on buoyant forces. One participant suggests integrating fluid pressure over the surface area to derive the buoyant force, while another proposes a simpler argument based on hydrostatic equilibrium. This argument posits that replacing a submerged object with an equal volume of fluid results in a net force equal to the weight of the displaced fluid, thus confirming Archimedes's principle. There are challenges in performing the integration correctly, with one participant expressing difficulty in their calculations. The conversation highlights the complexity of applying calculus to buoyancy problems, especially in varying gravitational fields.