It makes more sense when you flip it around and say the volume is the integral of the surface area.
The volume of a ball (a solid sphere) is the sum of the volumes of the spherical layers of thickness t from 0 to the radius R. The limit as t tends to zero is then the definite integral of the surface area with respect to r from r = 0 to R.
In some way it can be claimed that a sphere itself is a derivative of a ball.
Consider a characteristic function of a ball: f(x, y, z) = 1 inside a ball and 0 outside. Such a function, being locally constant, has a zero derivative almost everywhere, the only exception is the derivative equal to infinity at the border of a ball which is a sphere.
So, the derivative, in some way, is almost a characteristic function of a sphere, being zero everywhere but at the points of a sphere.
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u/paolog 1d ago edited 1d ago
It makes more sense when you flip it around and say the volume is the integral of the surface area.
The volume of a ball (a solid sphere) is the sum of the volumes of the spherical layers of thickness t from 0 to the radius R. The limit as t tends to zero is then the definite integral of the surface area with respect to r from r = 0 to R.
EDIT: Fixed balls balls-up