Confused about using the integral and infinitesimal?

[Dufoe]

I don't understand what is meant by "derive the formula for finding the volume of a sphere that uses infinitesimals but not the standard formula for the integral"?
Is this talking about Gauss or what? I'm completely self taught in calculus and I did three proofs already... the old cylinder / cone proof, and the other two used the standard formula for the integral.. Even any link to something that explains this would help. I'm only aware of 7 proofs and I don't get how any of them meet both criteria..? Thanks!

 

[DEvens]

Possibly the exact wording of the question would help.

 

[Hector Mata]

This may help:

http://math.stackexchange.com/quest...phere-by-dividing-it-into-infinitesimal-cubes

The idea is you already know the area of a cross-section of a sphere (a disk) from geometry: πr². So now, what happens if you try to stack disks on top of each other, making their height increasingly small (or, conversely, having a greater and greater number of thinner disks).

 

[leright]

Yup, as Hector is alluding to, you need to find the thickness of spherical shells. If you find the area of a shell and then multiply by a infinitessimal thickness you get the volume of the infinitessimally thick shell. Integrate these shell volumes to find the volume of the sphere. The idea to realize is that since the thickness of the shell is infinitessimal the shell area is the same in the inner and outer surface of the shell. as the thickness of the shell decreases the inner surface and outer surface areas get closer together. So for an infinitessimal thickness they are the same area.

 

[Svein]

I would start by observing that the volume of a sphere is twice the volume of half a sphere. This makes the formula for one "slice" much simpler.