Order of integration (calculus)

The problem for examination is evaluation of an integral of the form

:$\backslash iint\_D\; \backslash \; f(x,y\; )\; \backslash \; dx\; \backslash ,dy\; ,$

where D is some two-dimensional area in the xy–plane. For some functions f straightforward integration is feasible, but where that is not true, the integral can sometimes be reduced to simpler form by changing the order of integration. The difficulty with this interchange is determining the change in description of the domain D.

The method also is applicable to other multiple integrals.{{cite book |title=Multivariate Calculus and Geometry |author=Seán Dineen |page=162 |url=https://books.google.com/books?id=1YNX3YAf1vMC&dq=%22order+of+integration%22&pg=PA165

|isbn=1-85233-472-X |publisher=Springer |year=2001}}{{cite book |title=Introduction to Calculus and Analysis: Vol. II/1, II/2. Classics in mathematics |author= Richard Courant & Fritz John |url=https://books.google.com/books?id=ngkQxS4eicgC&dq=%22order+of+integration%22&pg=RA3-PA891

|page=897 |isbn=3-540-66569-2 |year=2000 |publisher=Springer }}

Sometimes, even though a full evaluation is difficult, or perhaps requires a numerical integration, a double integral can be reduced to a single integration, as illustrated next. Reduction to a single integration makes a numerical evaluation much easier and more efficient.

File:Integration Order.svg

Consider the iterated integral

:$\backslash int\_a^z\; \backslash ,\; \backslash int\_a^x\; \backslash ,\; h(y)\; \backslash ,\; dy\; \backslash ,\; dx\; ,$

In this expression, the second integral is calculated first with respect to y and x is held constant—a strip of width dx is integrated first over the y-direction (a strip of width dx in the x direction is integrated with respect to the y variable across the y direction), adding up an infinite amount of rectangles of width dy along the y-axis. This forms a three dimensional slice dx wide along the x-axis, from y=a to y=x along the y-axis, and in the z direction z=h(y). Notice that if the thickness dx is infinitesimal, x varies only infinitesimally on the slice. We can assume that x is constant.{{cite web |publisher=Department of Mathematics, Oregon State University |title=Double Integrals |date=1996 |url=https://math.oregonstate.edu/home/programs/undergrad/CalculusQuestStudyGuides/vcalc/255doub/255doub.html }} This integration is as shown in the left panel of Figure 1, but is inconvenient especially when the function h(y) is not easily integrated. The integral can be reduced to a single integration by reversing the order of integration as shown in the right panel of the figure. To accomplish this interchange of variables, the strip of width dy is first integrated from the line x = y to the limit x = z, and then the result is integrated from y = a to y = z, resulting in:

:$\backslash int\_a^z\; \backslash int\_a^x\; h(y)\; \backslash \; dy\backslash \; dx\; =\; \backslash int\_a^z\; h(y)\backslash \; dy\; \backslash \; \backslash int\_y^z\; dx\; =\; \backslash int\_a^z\; \backslash left(z-y\backslash right)\; h(y)\backslash ,\; dy\; .$

This result can be seen to be an example of the formula for integration by parts, as stated below:The prime " ′ " denotes a derivative in Lagrange's notation.

:$\backslash int\_a^z\; f(x)\; g\text{'}(x)\backslash ,\; dx\; =\; \backslash left[\; f(x)\; g(x)\; \backslash right]\_a^z\; -\; \backslash int\_a^z\; f\text{'}(x)\; g(x)\backslash ,\; dx$

Substitute:

:$f\; (x)\; =\; \backslash int\_a^x\; h(y)\backslash ,\; dy\; ~\backslash text\{\; and\; \}~\; g\text{'}(x)\; =\; 1\; .$

Which gives the result.

For application to principal-value integrals, see Whittaker and Watson,{{cite book |title=A Course of Modern Analysis: an introduction to the general theory of infinite processes and of analytic functions, with an account of the principal transcendental functions |author1=Edmund Taylor Whittaker|authorlink1=E. T. Whittaker|author2=George Neville Watson|authorlink2=G. N. Watson |page= §4.51, p. 75

|isbn=0-521-58807-3 |year=1927 |publisher=Cambridge University Press |edition=4th ed., repr }} Gakhov,{{cite book |title=Boundary Value Problems |page=46 |author=F. D. Gakhov |url=https://books.google.com/books?id=9G7sfwTDv8QC&dq=%22order+of+integration%22&pg=PA46 |isbn=0-486-66275-6 |publisher=Courier Dover Publications |year=1990}} Lu,{{cite book |title=Boundary Value Problems for Analytic Functions |author=Jian-Ke Lu |page= 44 |url=https://books.google.com/books?id=RFafUfgB1dAC&dq=principal+value+%22order+of++integration%22&pg=PA43

|isbn=981-02-1020-5 |year=1993 |publisher=World Scientific |location=Singapore }} or Zwillinger.{{cite book |title=Handbook of integration |author=Daniel Zwillinger |page=61 |url=https://books.google.com/books?id=DQd4wfV7fo0C&dq=%22order+of+integration%22&pg=PA61 |isbn=0-86720-293-9 |year=1992 |publisher=AK Peters Ltd.}} See also the discussion of the Poincaré-Bertrand transformation in Obolashvili.{{cite book |title=Higher order partial differential equations in Clifford analysis: effective solutions to problems |publisher=Birkhäuser |year=2003 |isbn=0-8176-4286-2 |author=Elena Irodionovna Obolashvili |url=https://books.google.com/books?id=HmvmB6NCyEAC&dq=principal+value+%22order+of++integration%22&pg=PA101

|page=101 }} An example where the order of integration cannot be exchanged is given by Kanwal:{{cite book |author= Ram P. Kanwal |title=Linear Integral Equations: theory and technique |page= 194 |url =https://books.google.com/books?id=-bV9Qn8NpCYC&dq=+%22Poincar%C3%A9-Bertrand+transformation%22&pg=PA194

|isbn=0-8176-3940-3 |year=1996 |publisher=Birkhäuser |location=Boston |edition=2nd}}

:$\backslash frac\; \{1\}\{(2\backslash pi\; i\; )^2\}\; \backslash int\_L^*\; \backslash frac\{d\{\backslash tau\}\_1\}\{\{\backslash tau\}\_1\; -\; t\}\backslash \; \backslash int\_L^*\backslash \; g(\backslash tau)\backslash frac\{d\; \backslash tau\}\{\backslash tau-\backslash tau\_1\}\; =\; \backslash frac\{1\}\{4\}\; g(t)\; \backslash \; ,$

while:

:$\backslash frac\; \{1\}\{(2\backslash pi\; i\; )^2\}\; \backslash int\_L^*\; g(\; \backslash tau\; )\; \backslash \; d\; \backslash tau\; \backslash left(\; \backslash int\_L^*\; \backslash frac\{d\; \backslash tau\_1\; \}\; \{\backslash left(\; \backslash tau\_1\; -\; t\backslash right)\; \backslash left(\; \backslash tau-\backslash tau\_1\; \backslash right)\}\; \backslash right)\; =\; 0\; \backslash \; .$

The second form is evaluated using a partial fraction expansion and an evaluation using the Sokhotski–Plemelj formula:For a discussion of the Sokhotski-Plemelj formula see, for example, {{cite book |title=The Cauchy Transform |author=Joseph A. Cima, Alec L. Matheson & William T. Ross |page= 56 |url=https://books.google.com/books?id=1sVLg512ffIC&dq=%22Plemelj+formula%22&pg=PA56

|isbn=0-8218-3871-7 |year=2006 |publisher=American Mathematical Society }} or {{cite book |title=Linear integral equations |author=Rainer Kress |page= Theorem 7.6, p. 101 |url=https://books.google.com/books?id=R3BIOfKssQ4C&dq=%22Plemelj+formula%22&pg=PA115

|isbn=0-387-98700-2 |year=1999 |publisher=Springer |edition=2nd }}

:$\backslash int\_L^*\backslash frac\{d\; \backslash tau\_1\}\{\backslash tau\_1-t\}\; =\; \backslash int\_L^*\; \backslash frac\; \{d\backslash tau\_1\}\{\backslash tau\_1-t\}\; =\; \backslash pi\backslash \; i\; \backslash \; .$

The notation $\backslash int\_L^*$ indicates a Cauchy principal value. See Kanwal.

A discussion of the basis for reversing the order of integration is found in the book Fourier Analysis by T.W. Körner.{{cite book |title=Fourier Analysis |author=Thomas William Körner |page=Chapters 47 & 48 |url=https://books.google.com/books?id=DZTDtXs4OQAC&q=Fourier+analysis+subject:%22Fourier+analysis%22

|isbn=0-521-38991-7 |publisher=Cambridge University Press |year=1988 }} He introduces his discussion with an example where interchange of integration leads to two different answers because the conditions of Theorem II below are not satisfied. Here is the example:

:$\backslash int\_1^\{\backslash infty\}\; \backslash frac\; \{x^2-y^2\}\{\backslash left(x^2+y^2\backslash right)^2\}\backslash \; dy\; =\; \backslash left[\backslash frac\{y\}\{x^2+y^2\}\backslash right]\_1^\{\backslash infty\}\; =\; -\backslash frac\{1\}\{1+x^2\}\; \backslash \; \backslash left[x\; \backslash ge\; 1\; \backslash right]\backslash \; .$

:::$\backslash int\_1^\{\backslash infty\}\; \backslash left(\; \backslash int\_1^\{\backslash infty\}\backslash frac\; \{x^2-y^2\}\{\backslash left(x^2+y^2\backslash right)^2\}\backslash \; dy\; \backslash right)\backslash \; dx\; =\; -\backslash frac\{\backslash pi\}\{4\}\; \backslash \; .$

:::$\backslash int\_1^\{\backslash infty\}\; \backslash left(\; \backslash int\_1^\{\backslash infty\}\backslash frac\; \{x^2-y^2\}\{\backslash left(x^2+y^2\backslash right)^2\}\backslash \; dx\; \backslash right)\backslash \; dy\; =\; \backslash frac\{\backslash pi\}\{4\}\; \backslash \; .$

Two basic theorems governing admissibility of the interchange are quoted below from Chaudhry and Zubair:{{cite book |title=On a Class of Incomplete Gamma Functions with Applications |author=M. Aslam Chaudhry & Syed M. Zubair |page=Appendix C |url=https://books.google.com/books?id=Edf4KrG_vlYC&dq=%22order+of+integration%22&pg=PA458 |isbn=1-58488-143-7 |publisher=CRC Press |year=2001}}

{{math_theorem

|name=Theorem I

|math_statement= Let f(x, y) be a continuous function of constant sign defined for a ≤ x < ∞, c ≤ y < ∞, and let the integrals

{{Center|$J(y):=\; \backslash int\_a^\backslash infty\; f(x,\backslash \; y)\; \backslash ,\; dx${{space|10}} and {{space|10}}$J^*(x)\; =\; \backslash int\_c^\backslash infty\; f(x,\; \backslash \; y)\; \backslash ,\; dy$}} regarded as functions of the corresponding parameter be, respectively, continuous for c ≤ y < ∞, a ≤ x < ∞. Then if at least one of the iterated integrals

{{Center|$\backslash int\_c^\backslash infty\; \backslash left(\backslash int\_a^\backslash infty\; \backslash \; f(x,\backslash \; y)\; dx\backslash right\; )\; dy${{space|10}} and {{space|10}}$\backslash int\_a^\backslash infty\; \backslash \; \backslash left(\backslash int\_c^\backslash infty\; \backslash \; f(x,\backslash \; y)\; dy\backslash right\; )\; dx$}} converges, the other integral also converges and their values coincide.

}}

{{math_theorem

|name=Theorem II

|math_statement= Let f(x, y) be continuous for a ≤ x < ∞, c ≤ y < ∞, and let the integrals

{{Center|$J(y):=\; \backslash int\_a^\backslash infty\; f(x,\backslash \; y)\backslash ,\; dx${{space|10}} and {{space|10}}$J^*(x)\; =\; \backslash int\_c^\backslash infty\; f(x,\; \backslash \; y)\; \backslash ,\; dy$}} be respectively, uniformly convergent on every finite interval c ≤ y < C and on every finite interval a ≤ x < A. Then if at least one of the iterated integrals

{{Center|$\backslash int\_c^\backslash infty\; \backslash left(\; \backslash int\_a^\backslash infty\; |f(x,\backslash \; y)|\; dx\; \backslash right)\; dy${{space|10}} and {{space|10}}$\backslash int\_a^\backslash infty\; \backslash \; \backslash left(\backslash int\_c^\backslash infty\; |f(x,\backslash \; y)|\; dy\; \backslash right\; )\; dx$}} converges, the iterated integrals

{{Center|$\backslash int\_c^\backslash infty\; \backslash left(\; \backslash int\_a^\{\backslash infty\}\; f(x,\backslash \; y)\; dx\; \backslash right)\; dy${{space|10}} and {{space|10}}$\backslash int\_a^\backslash infty\; \backslash left(\backslash int\_c^\backslash infty\; f(x,\backslash \; y)\; dy\backslash right)\; dx$}} also converge and their values are equal.

}}

The most important theorem for the applications is quoted from Protter and Morrey:{{cite book |title=Intermediate Calculus |author=Murray H. Protter & Charles B. Morrey, Jr. |page=307 |url=https://books.google.com/books?id=AXw4a2_vzt4C&pg=PA307 |isbn=0-387-96058-9 |publisher=Springer |year=1985}}

{{math_theorem

|name=

|math_statement=Suppose F is a region given by $F=\backslash left\backslash \{(x,\backslash \; y):a\; \backslash le\; x\; \backslash le\; b,\; p(x)\; \backslash le\; y\; \backslash le\; q(x)\; \backslash right\backslash \}\; \backslash ,$  where p and q are continuous and p(x) ≤ q(x) for a ≤ x ≤ b. Suppose that f(x, y) is continuous on F. Then

{{Center|$\backslash iint\_F\; f(x,y)\; \backslash ,dA\; =\; \backslash int\_a^b\; \backslash int\_\{p(x)\}^\{q(x)\}\; f(x,\backslash \; y)\backslash ,dy\backslash \; dx\; .$}} The corresponding result holds if the closed region F has the representation $F=\backslash left\backslash \{(x,\backslash \; y):c\backslash le\; y\; \backslash le\; d,\backslash \; r(y)\; \backslash le\; x\; \backslash le\; s(y)\backslash right\backslash \}$  where r(y) ≤ s(y) for c ≤ y ≤ d.  In such a case,

: $\backslash iint\_F\; f(x,\backslash \; y)\; dA\; =\; \backslash int\_c^d\; \backslash int\_\{r(y)\}^\{s(y)\}\; f(x,\backslash \; y)\backslash ,\; dx\backslash \; dy\; \backslash \; .$

In other words, both iterated integrals, when computable, are equal to the double integral and therefore equal to each other.

}}

• Fubini's theorem

{{reflist|2}}

• [http://tutorial.math.lamar.edu/Classes/CalcIII/DIGeneralRegion.aspx Paul's Online Math Notes: Calculus III]
• [https://math.oregonstate.edu/home/programs/undergrad/CalculusQuestStudyGuides/vcalc/255doub/255doub.html Good 3D images showing the computation of "Double Integrals" using iterated integrals], the Department of Mathematics at Oregon State University.
• [https://www.math.ucla.edu/~ronmiech/Calculus_Problems/32B/chap13/section3/ Ron Miech's UCLA Calculus Problems] More complex examples of changing the order of integration (see Problems 33, 35, 37, 39, 41 & 43)
• [http://www.math.umn.edu/~nykamp/m2374/readings/doubleintchange/ Duane Nykamp's University of Minnesota website]

{{Integrals}}

Category:Integral calculus