Math 2.0

Measure and integration Proposition

Volume of a box

Builds on Jordan measureTriple integralFubini's theorem

Statement

Let ai<bia_i < b_i for i=1,…,ni = 1, \dots, n, and let Q=[a1,b1]×⋯×[an,bn]⊆ℝnQ = [a_1, b_1] \times \dots \times [a_n, b_n] \subseteq \mathbb{R}^n. Then QQ is Jordan measurable, and its volume is vol⁡(Q)=∫Q1 dV=(b1−a1)(b2−a2)⋯(bn−an).\operatorname{vol}(Q) = \int_Q 1 \, dV = (b_1 - a_1)(b_2 - a_2) \cdots (b_n - a_n).

Remarks

For a box, the Jordan measure and the Riemann integral of the constant function 11 give the same value. The integral can be computed with Fubini's theorem as an iterated integral of 11, one factor bi−aib_i - a_i at a time. Open and half-open boxes with the same edges have the same volume, because the faces of QQ have volume zero. For n=3n = 3 this is the formula length times width times height, a case of base area times height. The volume of a general solid EE is the triple integral ∭E1 dV\iiint_E 1 \, dV.

Reading path

To understand this page from the ground up, read these 13 pages in order. The foundations come first.

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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 integraldefinitionVolume of a box, statement, no evidence levelVolume of a boxstatement
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. 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. Volume of a box statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Volume, Basic shapes
  2. Web https://en.wikipedia.org/wiki/Volume, Integral calculus
About this page
Tag0VEKNG02W0
URIm2://0VEKNG02W0
Aliasesvolume-of-a-rectangular-box
Kindstatement, proposition
WritingW0 stub
MSC28A75
StatusThis copy of the site has no build data. Run m2 status 0VEKNG02W0.
UsesJordan measure, Triple integral, Fubini's theorem

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