Integral Calculus, Atmospheric Pressure

Atmospheric Pressure

At any given altitude, the atmospheric pressure is proportional to the amount of air above you. This in turn is the integral of the density of the air at all layers above. Assuming a constant temperature, pressure and density are proportional. Thus the pressure is the integral of the pressure at the higher levels.

Let p(x) be the atmospheric pressure, starting at x = 0 (sea level). Write p(x) as c times the integral of p(t) as t runs from x to ∞. Here c is an appropriate constant of proportionality. Differentiate to show p′(x) = -cp(x). Thus the change in pressure is proportional to the pressure. It must be an exponential function; p(x) = kE^-cx, where k and c are constants.

Unlike air, water is virtually incompressible. The density does not change as you descend to the bottom of the ocean. Hence the pressure is proportional to the depth.