Math 2.0

Real functions Proposition

Changing the order of integration

Builds on Fubini's theoremIterated integralIntegration over a non-rectangular region

Statement

Let D⊆ℝ2D \subseteq \mathbb{R}^2 be a region that has two descriptions, D={(x,y):a≤x≤b, φ1(x)≤y≤φ2(x)}={(x,y):c≤y≤d, ψ1(y)≤x≤ψ2(y)},D = \{(x, y) : a \le x \le b, \ \varphi_1(x) \le y \le \varphi_2(x)\} = \{(x, y) : c \le y \le d, \ \psi_1(y) \le x \le \psi_2(y)\}, where φ1≤φ2\varphi_1 \le \varphi_2 are continuous on [a,b][a, b] and ψ1≤ψ2\psi_1 \le \psi_2 are continuous on [c,d][c, d]. Let f ⁣:D→ℝf \colon D \to \mathbb{R} be continuous. Then ∫ab∫φ1(x)φ2(x)f(x,y) dy dx=∫cd∫ψ1(y)ψ2(y)f(x,y) dx dy,\int_a^b \int_{\varphi_1(x)}^{\varphi_2(x)} f(x, y) \, dy \, dx = \int_c^d \int_{\psi_1(y)}^{\psi_2(y)} f(x, y) \, dx \, dy, and both equal the double integral of ff over DD.

Remarks

The proof applies Fubini's theorem on a rectangle that contains DD to ff times the indicator function of DD. The main work is to describe DD again in the other order. One way is to sketch the region. Example: ∫01∫x1ey2 dy dx\int_0^1 \int_x^1 e^{y^2} \, dy \, dx has no elementary inner antiderivative. The region is 0≤x≤y≤10 \le x \le y \le 1, so the integral equals ∫01∫0yey2 dx dy=∫01yey2 dy=(e−1)/2\int_0^1 \int_0^y e^{y^2} \, dx \, dy = \int_0^1 y e^{y^2} \, dy = (e - 1)/2. For unbounded regions or discontinuous integrands, a change of order needs extra hypotheses.

Reading path

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

Show the path as a graph
Euclidean space, object, no evidence levelEuclidean spaceobjectFunction, concept, no evidence levelFunctionconceptFunction of several variables, concept, no evidence levelFunction of several v…conceptDouble integral, concept, no evidence levelDouble integralconceptIterated integral, definition, no evidence levelIterated integraldefinitionIterated integral, concept, no evidence levelIterated integralconceptLimit of a function of several variables, concept, no evidence levelLimit of a function o…conceptJordan measure, concept, no evidence levelJordan measureconceptContinuity in R^n, definition, no evidence levelContinuity in R^ndefinitionDouble integral, definition, no evidence levelDouble integraldefinitionIntegration over a non-rectangular region, concept, no evidence levelIntegration over a no…conceptFubini's theorem, statement, no evidence levelFubini's theoremstatementChanging the order of integration, statement, no evidence levelChanging the order of…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 of several variables concept
  4. Double integral concept
  5. Iterated integral definition
  6. Iterated integral concept
  7. Limit of a function of several variables concept
  8. Jordan measure concept
  9. Continuity in R^n definition
  10. Double integral definition
  11. Integration over a non-rectangular region concept
  12. Fubini's theorem statement
  13. Changing the order of integration statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Order_of_integration_(calculus), Problem statement
  2. Web https://en.wikipedia.org/wiki/Multiple_integral, Double integral over a normal domain
About this page
Tag0H60H1Q0SX
URIm2://0H60H1Q0SX
Aliasesreversing-order-of-integration
Kindstatement, proposition
WritingW0 stub
MSC26B15
StatusThis copy of the site has no build data. Run m2 status 0H60H1Q0SX.
UsesFubini's theorem, Iterated integral, Integration over a non-rectangular region
Same as

Markdown · JSON · Edit on GitHub

What this page needs for W1

  • an inventory row: the row of the inventory that the node accounts for (0 of 1)

The writing levels say how complete the text of a page is. They do not measure its truth.

What links here

No page links to this page yet.