Suppose that the density of seawater, p = p (z), varies with the depth below the surface.

Question:

Suppose that the density of seawater, p = p (z), varies with the depth below the surface.
(a) Show that the hydrostatic pressure is governed by the differential equation

where is the acceleration due to gravity. Let Po and po be the pressure and density at z = 0. Express the pressure at depth as an integral.
(b) Suppose the density of seawater at depth is given by p = p0ez/H, where H is a positive constant. Find the total force, expressed as an integral, exerted on a vertical circular porthole of radius whose center is located at a distance L > r below the surface.