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Q: Is it a coincidence that a circles circumference is the derivative of its area, as well as the volume of a sphere being the antiderivative of its surface area? What is the explanation for this?

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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.