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Slicing gives the unit-ball volumes through dimension five
Example
Writing , slicing gives
Facts & Assumptions
Given: Unit closed balls in positive integer dimensions.
The Wallis integrals satisfy , , and for (Wallis integrals satisfy the two-step recurrence, closed forms, and the adjacent-integral squeeze).
Verification
Starting with , [F1] and the upper-semicircle area give , while gives .
With , by [F2], giving . Also , giving .
Combining the preceding calculations gives the displayed table without using the Gamma closed form.
Depends on
- Closed Euclidean balls are Jordan measurable and their volumes satisfy the slicing recursion
- A closed disc of radius $r\ge0$ has Jordan content $\pi r^2$
- Riemann area between continuous graphs equals Jordan content
- Wallis integrals satisfy the two-step recurrence, closed forms, and the adjacent-integral squeeze
- For a natural $n \ge 1$ the function $x \mapsto x^{n}$ is differentiable everywhere with derivative $\iota(n)\,x^{\,n-1}$; for $n = 0$ it is the constant $1$, with derivative $0$; for a natural $n \ge 1$ the function $x \mapsto x^{-n}$ is differentiable at every $x \ne 0$ with derivative $-\iota(n)\,x^{-n-1}$; consequently every polynomial function is differentiable at every real, with the derivative computed term by term
- The second fundamental theorem: if $G$ is differentiable on $[a,b]$ with $G' = f$ and $f$ is integrable, then $\int_a^b f = G(b)-G(a)$
Used by
Nothing in the library uses this result yet.
Dependency tree · two levels
53 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
- Sheldon Axler, Measure, Integration & Real Analysis, §5C (standard reference, not scraped)
- University of Toronto MAT237Y1, The Gamma Function and the Beta Function, §2.4 (standard reference, not scraped)