Bouyant Force on submerged hollow object with air trapped inside

Hello,

I'm reading about buoyant forces, but my book does not mention anything about an object with air (or any gas/fluid whose density is less than that of the fluid the whole object is immersed in). So for simplicity's sake, if you have a sealed, hollow cylinder completely submerged in water with air trapped inside, how do you calculate the buoyant force(s)? I was thinking maybe it was

F_B,air = ρ_airgV_air
F_B,water = ρ_watergV_water

then add them together but that doesn't seem right...