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I was asked to explain why the volume of a sphere is $\\frac{4}{3}\\pi r^3$ to a student that does not have the knowledge of calculus The subtle question here is why the residual volume (between the volume of the sphere and the toothed solid obtained as the infinite union of infinitely thin disks) can be neglected, whereas in evaluating the area of the surface, for example, such an approximation is inadequate. In doing so i thought of an argument and i cannot seem to find t.

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Now if i have a sphere of radius r, and i increase the radius by a tiny amount, dr, then the new, expanded sphere has a volume that is bigger, by the volume of the thin spherical shell that was just added. Second, how do i mathematically limit the volume of the cylinder to be less than that of a sphere The volume of sphere using integrals ask question asked 8 years, 2 months ago modified 1 year, 7 months ago

How proof that $\omega$ is the volume form

I want a simple proof for the formula of volume of sphere Does the proof/explanation without integration possible? Rate of change of volume in a sphere ask question asked 9 years, 10 months ago modified 6 years, 5 months ago Volume form on a sphere

Ask question asked 11 years, 11 months ago modified 8 years, 10 months ago 3 the first question that comes into my mind here is whether any cylinder that touches (at 4 pts) the circumference of the sphere and does not go out of it, has equal volume