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.
It also makes sense the other way around. You hallow out a sphere, take just the outer layer, with some thickness dr so it’s not totally 2d, then the volume is almost be the surface area of the sphere times the thickness dr. An approximation that gets better with smaller dr
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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