Math 2.0

Measure and integration Product measures Theorem

Cavalieri's principle

Builds on Fubini's theoremJordan measureTriple integral

Statement

Let E⊆ℝ3E \subseteq \mathbb{R}^3 be a Jordan measurable solid that lies between the planes z=az = a and z=bz = b. For z∈[a,b]z \in [a, b], let Ez={(x,y):(x,y,z)∈E}E_z = \{(x, y) : (x, y, z) \in E\} be its cross-section at height zz, and suppose that each EzE_z is Jordan measurable with area A(z)A(z). Then vol⁡(E)=∫abA(z) dz.\operatorname{vol}(E) = \int_a^b A(z) \, dz. In particular, if two such solids have cross-sections of equal area at each height zz, then they have equal volume.

Remarks

This follows from Fubini's theorem applied to the indicator function of EE. Example: the ball of radius rr has cross-sections of area π(r2−z2)\pi (r^2 - z^2). A cylinder of radius rr and height rr with a cone removed has the same cross-sections, which gives the volume 23πr3\tfrac{2}{3} \pi r^3 for a half ball. The plane version says that two regions with equal chord lengths on each line parallel to a fixed line have equal area. Archimedes used a similar method in the 3rd century BC. Cavalieri developed it as the method of indivisibles in 1635.

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Euclidean space, object, no evidence levelEuclidean spaceobjectFunction, concept, no evidence levelFunctionconceptInfimum, concept, no evidence levelInfimumconceptSupremum, concept, no evidence levelSupremumconceptFunction of several variables, concept, no evidence levelFunction of several v…conceptJordan measure, definition, no evidence levelJordan measuredefinitionIterated integral, definition, no evidence levelIterated integraldefinitionLimit of a function of several variables, concept, no evidence levelLimit of a function o…conceptContinuity in R^n, definition, no evidence levelContinuity in R^ndefinitionDouble integral, definition, no evidence levelDouble integraldefinitionFubini's theorem, statement, no evidence levelFubini's theoremstatementTriple integral, definition, no evidence levelTriple integraldefinitionCavalieri's principle, statement, no evidence levelCavalieri's principlestatement
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  1. Euclidean space object
  2. Function concept
  3. Infimum concept
  4. Supremum concept
  5. Function of several variables concept
  6. Jordan measure definition
  7. Iterated integral definition
  8. Limit of a function of several variables concept
  9. Continuity in R^n definition
  10. Double integral definition
  11. Fubini's theorem statement
  12. Triple integral definition
  13. Cavalieri's principle statement

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Further reading

  1. Web https://en.wikipedia.org/wiki/Cavalieri%27s_principle, 3-dimensional
  2. Web https://en.wikipedia.org/wiki/Cavalieri%27s_principle, Generalisation to measures
  3. Web https://en.wikipedia.org/wiki/Cavalieri%27s_principle, History
About this page
Tag0GTZ2F3PQY
URIm2://0GTZ2F3PQY
Aliasesmethod-of-indivisibles
Kindstatement, theorem
WritingW1 stated
MSC28A75
StatusThis copy of the site has no build data. Run m2 status 0GTZ2F3PQY.
UsesFubini's theorem, Jordan measure, Triple integral
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