Math 2.0

Real functions Proposition

Volume of a ball in R^3

Builds on Jacobian of spherical coordinatesVolume of a ball in R^nCavalieri's principle

Statement

Let R>0R > 0, and let B={(x,y,z)∈ℝ3:x2+y2+z2≤R2}B = \{(x, y, z) \in \mathbb{R}^3 : x^2 + y^2 + z^2 \le R^2\}. Then vol⁡(B)=43πR3.\operatorname{vol}(B) = \frac{4}{3} \pi R^3.

Remarks

In spherical coordinates, vol⁡(B)=∫02π∫0π∫0Rρ2sin⁡φ dρ dφ dθ=2π⋅2⋅R3/3\operatorname{vol}(B) = \int_0^{2\pi} \int_0^\pi \int_0^R \rho^2 \sin \varphi \, d\rho \, d\varphi \, d\theta = 2\pi \cdot 2 \cdot R^3 / 3. By slices, the cross-section at height zz is a disc of area π(R2−z2)\pi (R^2 - z^2), and ∫−RRπ(R2−z2) dz=43πR3\int_{-R}^{R} \pi (R^2 - z^2) \, dz = \tfrac{4}{3} \pi R^3. This is Cavalieri's principle. This is the case n=3n = 3 of the volume of a ball in R^n, since Γ(5/2)=34π\Gamma(5/2) = \tfrac{3}{4} \sqrt{\pi}. Archimedes found this volume, by comparison with a cone and a cylinder.

Reading path

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Euclidean space, object, no evidence levelEuclidean spaceobjectFunction, concept, no evidence levelFunctionconceptFunction composition, concept, no evidence levelFunction compositionconceptInfimum, concept, no evidence levelInfimumconceptSupremum, concept, no evidence levelSupremumconceptFréchet derivative, definition, no evidence levelFréchet derivativedefinitionFunction of several variables, concept, no evidence levelFunction of several v…conceptJordan measure, definition, no evidence levelJordan measuredefinitionLimit of a function of several variables, concept, no evidence levelLimit of a function o…conceptIterated integral, definition, no evidence levelIterated integraldefinitionPartial derivative, concept, no evidence levelPartial derivativeconceptContinuity in R^n, definition, no evidence levelContinuity in R^ndefinitionDifferentiable function of several variables, definition, no evidence levelDifferentiable functi…definitionPartial derivative, definition, no evidence levelPartial derivativedefinitionDouble integral, definition, no evidence levelDouble integraldefinitionChain rule in several variables, statement, no evidence levelChain rule in several…statementDifferentiable on an open set, definition, no evidence levelDifferentiable on an …definitionJacobian matrix, definition, no evidence levelJacobian matrixdefinitionHigher partial derivatives, definition, no evidence levelHigher partial deriva…definitionFubini's theorem, statement, no evidence levelFubini's theoremstatementTriple integral, definition, no evidence levelTriple integraldefinitionDerivative of the inverse function, statement, no evidence levelDerivative of the inv…statementIterated derivative, definition, no evidence levelIterated derivativedefinitionJacobian determinant, definition, no evidence levelJacobian determinantdefinitionCavalieri's principle, statement, no evidence levelCavalieri's principlestatementC^k function characterised by continuous derivatives, statement, no evidence levelC^k function characte…statementInverse function theorem, statement, no evidence levelInverse function theo…statementDiffeomorphism, definition, no evidence levelDiffeomorphismdefinitionPolar coordinates, definition, no evidence levelPolar coordinatesdefinitionChange of variables in multiple integrals, statement, no evidence levelChange of variables i…statementSpherical coordinates, definition, no evidence levelSpherical coordinatesdefinitionVolume of a ball in R^n, statement, no evidence levelVolume of a ball in R…statementJacobian of spherical coordinates, statement, no evidence levelJacobian of spherical…statementVolume of a ball in R^3, statement, no evidence levelVolume of a ball in R…statement
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Euclidean space object
  2. Function concept
  3. Function composition concept
  4. Infimum concept
  5. Supremum concept
  6. Fréchet derivative definition
  7. Function of several variables concept
  8. Jordan measure definition
  9. Limit of a function of several variables concept
  10. Iterated integral definition
  11. Partial derivative concept
  12. Continuity in R^n definition
  13. Differentiable function of several variables definition
  14. Partial derivative definition
  15. Double integral definition
  16. Chain rule in several variables statement
  17. Differentiable on an open set definition
  18. Jacobian matrix definition
  19. Higher partial derivatives definition
  20. Fubini's theorem statement
  21. Triple integral definition
  22. Derivative of the inverse function statement
  23. Iterated derivative definition
  24. Jacobian determinant definition
  25. Cavalieri's principle statement
  26. C^k function characterised by continuous derivatives statement
  27. Inverse function theorem statement
  28. Diffeomorphism definition
  29. Polar coordinates definition
  30. Change of variables in multiple integrals statement
  31. Spherical coordinates definition
  32. Volume of a ball in R^n statement
  33. Jacobian of spherical coordinates statement
  34. Volume of a ball in R^3 statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Ball_(mathematics), Volume
  2. Web https://en.wikipedia.org/wiki/Multiple_integral, Spherical coordinates
  3. Web https://en.wikipedia.org/wiki/Cavalieri%27s_principle, History
About this page
Tag0WMDV48VSN
URIm2://0WMDV48VSN
Aliasesvolume-of-a-sphere
Kindstatement, proposition
WritingW0 stub
MSC26B15
StatusThis copy of the site has no build data. Run m2 status 0WMDV48VSN.
UsesJacobian of spherical coordinates, Volume of a ball in R^n, Cavalieri's principle

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