A solid occupies a region E with surface S and is immersed in a liquid with constant density $.$ We set up a coordinate system so that the xy$-$plane coincides with the surface of the liquid, and positive values of z are measured downward into the liquid. Then the pressure at depth z is $$p$=$gz$,$ where g is the acceleration due to gravity.$$ Explain why the total buoyant force on the solid due to the pressure distribution is given by the surface integral

F$=$Sp$($n$^)$dS

where n$^$ is the outer unit normal. Hint: this is similar to our calculation of the total gravitational force exerted by a spherical shell.