Alphabeta MathHow statement and proof provenance work
The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.
- Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
- AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
- AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.
These labels describe origin, not correctness: citations and verification chips remain separate evidence.
Solids of revolution about a coordinate axis
Definition
Let and let be continuous. The solid obtained by revolving the profile about the -axis is
If instead , the solid obtained by revolving the region under about the -axis is
The first description has perpendicular disc sections. The second has cylindrical shells. Zero values of , a zero inner radius, and the degenerate interval are included.
Depends on
- Euclidean spheres and closed balls as subspaces of $\mathbb{R}^n$
- Continuity of $f : A \to \mathbb{R}$ at a point of $A$ and on $A$: the $\varepsilon$-$\delta$ condition, its agreement with $\lim_{x \to c} f(x) = f(c)$ at a limit point, and continuity at an isolated point
- Intervals of $\mathbb{R}$: the nine order-convex forms, nondegeneracy, and length
Used by
Dependency tree · two levels
16 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.
Sources
- Sigurd Angenent, Math 221 lecture notes, Chapter 8 §§3–5 (standard reference, not scraped)