The volume of a sphere is , and the surface area is , which is again the derivative. If you describe volume, V, in terms of the radius, R, then increasing R will result in an increase in V that’s proportional to the surface area. If the surface area is given by S(R), then you’ll find that for a tiny change in the radius, dR, , or . This same argument can be used to show that the volume is the integral of the surface area (just keep painting layer after layer). The change in area, dA, is dA = (2πR)dR.

When you want to outsmart the world, you turn to the facts. And the facts are in the science.