Derivation of archimedes principleCan you pease tell me

Proof of Archimedes principleConsider a uniform cylinder immersed in a liquid as shown in Figure 3.Force on the upper face of the cylinder = hρgAForce on the lower face of the cylinder = [h + L]ρgADifference in force = LρgABut LA is the volume of liquid displaced by the cylinder, and LrgA is the weight of the liquid displaced by the cylinder.Therefore there is a net upward force on the cylinder equal to the weight of the fluid displaced by it.The same result will be obtained for a body of any shape, regular or not by taking into account the vertical and horirontal components of the forces on the object.If a sphere of radius r made of material of density σ is fully immersed in a liquid of density ρ the apparent weight of the sphere is given by:Apparent weight = actual weight - upthrust = 4/3 πr3g(ρ - s)The fluid may be a liquid, such as water, or a gas, such as air, although due to the low density of air (about 1 kg m3) the upthrust in air in usually small, but sufficient to support a helium-filled balloon or a hot air balloon.