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| {{for|the summary statistic in time series|Order of integration}}
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| {{Calculus |Integral}}
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| In [[calculus]], interchange of the '''order of integration''' is a methodology that transforms [[iterated integral]]s (or [[multiple integrals]] through the use of [[Fubini's theorem]]) of functions into other, hopefully simpler, integrals by changing the order in which the integrations are performed. In some cases, the order of integration can be validly interchanged; in others it cannot.
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| == Problem statement ==
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| The problem for examination is evaluation of an integral of the form
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| :<math> \iint_D \ f(x,y ) \ dx \,dy , </math>
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| 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''.
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| The method also is applicable to [[multiple integrals]].<ref name=Dineen>{{cite book |title=Multivariate Calculus and Geometry |author=Seán Dineen |page=162 |url=http://books.google.com/books?id=1YNX3YAf1vMC&pg=PA165&dq=%22order+of+integration%22&lr=&as_brr=0&sig=hF51z9mPjvfq_lisOmHwuLUrM0Q#PPA162,M1
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| |isbn=1-85233-472-X |publisher=Springer |year=2001}}</ref><ref name=Courant>{{cite book |title=Introduction to Calculus and Analysis: Vol. II/1, II/2. Classics in mathematics |author= Richard Courant & Fritz John |url=http://books.google.com/books?id=ngkQxS4eicgC&pg=RA3-PA891&dq=%22order+of+integration%22&lr=&as_brr=0&sig=EKZQrRpPezYV_67Bj71WN8awPyg#PRA3-PA897,M1
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| |page=897 |isbn=3-540-66569-2 |year=2000 |publisher=Springer }}</ref>
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| 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 integration|numerical evaluation]] much easier and more efficient.
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| ==Relation to integration by parts==
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| [[File:Integration Order.PNG|thumb |300px|left |Figure 1: Integration over the triangular area can be done using vertical or horizontal strips as the first step. This is an overhead view, looking down the z-axis onto the x-y plane. The sloped line is the curve ''y = x''.]]
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| Consider the iterated integral
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| :<math> \int_a^z \, \int_a^x \, h(y) \, dy \, dx \ </math>,
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| which we will write using the prefix notation commonly seen in physics:
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| :<math> \int_a^z dx \, \int_a^x \, h(y) \, dy </math>.
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| 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=f(x,y). Notice that if the thickness dx is infinitesimal, x varies only infinitesimally on the slice. We can assume that x is constant.<ref name=OSU>{{cite web|author= Department of Mathematics, Oregon State University |title=Double Integrals |url =http://www.math.oregonstate.edu/home/programs/undergrad/CalculusQuestStudyGuides/vcalc/255doub/255doub.html}}</ref> 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:
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| :<math> \int_a^z \ dx\ \int_a^x \ h(y) \ dy \ = \int_a^z \ h(y)\ dy \ \ \int_y^z \ dx = \int_a^z \ \left(z-y\right) h(y)\, dy \ .</math>
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| This result can be seen to be an example of the formula for [[integration by parts]], as stated below:<ref>The ''prime'' notation "''' <nowiki>′</nowiki> '''" denotes a derivative.</ref>
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| :<math>\int_a^z f(x) g'(x)\, dx = \left[ f(x) g(x) \right]_a^z - \int_a^z f'(x) g(x)\, dx\!</math>
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| Substitute:
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| :<math> f (x) = \int_a^x \ h(y)\, dy \text{ and }g'(x) = 1 \ . </math>
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| Which gives the result.
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| ==Principal-value integrals==
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| For application to [[Cauchy principal value|principal-value integrals]], see Whittaker and Watson,<ref name=Whittaker>{{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 |author=Edmund Taylor Whittaker & George Neville Watson |page= §4.51, p. 75 |url=http://books.google.com/books?id=ULVdGZmi9VcC&pg=PA75&dq=principal+value+%22order+of++integration%22&lr=&as_brr=0&sig=bx7JymRGBqKQ-K9KHRwRJk8XoYI
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| |isbn=0-521-58807-3 |year=1927 |publisher=Cambridge University Press |edition=4th ed., repr }}</ref> Gakhov,<ref name=Gakhov>{{cite book |title=Boundary Value Problems |page=46 |author=F. D. Gakhov |url=http://books.google.com/books?id=9G7sfwTDv8QC&pg=PA46&dq=%22order+of+integration%22&lr=&as_brr=0&sig=jw8C_74EzuwTmHXOAPRX9KrNi5g |isbn=0-486-66275-6 |publisher=Courier Dover Publications |year=1990}}</ref> Lu,<ref name=Lu>{{cite book |title=Boundary Value Problems for Analytic Functions |author=Jian-Ke Lu |page= 44 |url=http://books.google.com/books?id=RFafUfgB1dAC&pg=PA43&dq=principal+value+%22order+of++integration%22&lr=&as_brr=0&sig=IpfqQYcG__ljaZWIAqDy9Uji4e4#PPA44,M1
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| |isbn=981-02-1020-5 |year=1993 |publisher=World Scientific |location=Singapore }}</ref> or Zwillinger.<ref name=Zwillinger>{{cite book |title=Handbook of integration |author=Daniel Zwillinger |page=61 |url=http://books.google.com/books?id=DQd4wfV7fo0C&pg=PA61&dq=%22order+of+integration%22&lr=&as_brr=0&sig=pev_kroO46LEqZcxY1qJA01qv9U |isbn=0-86720-293-9 |year=1992 |publisher=AK Peters Ltd.}}</ref> See also the discussion of the Poincaré-Bertrand transformation in Obolashvili.<ref name= 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=http://books.google.com/books?id=HmvmB6NCyEAC&pg=PA101&dq=principal+value+%22order+of++integration%22&lr=&as_brr=0&sig=1ZcvrKq8hD6Zf-6PL2Bo6H0KEp0
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| |page=101 }}</ref> An example where the order of integration cannot be exchanged is given by Kanwal:<ref name=Kanwal>{{cite book |author= Ram P. Kanwal |title=Linear Integral Equations: theory and technique |page= 194 |url =http://books.google.com/books?id=-bV9Qn8NpCYC&pg=PA194&lpg=PA194&dq=+%22Poincar%C3%A9-Bertrand+transformation%22&source=web&ots=iofB7oQccG&sig=2yieQ-eUpZTZtPcZrJJpBZAO-R4&hl=en#PPA194,M1
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| |isbn=0-8176-3940-3 |year=1996 |publisher=Birkhäuser |location=Boston |edition=2nd Edition}}</ref>
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| :<math>\frac {1}{(2\pi i )^2} \int_L^* \frac{d{\tau}_1}{{\tau}_1 - t}\ \int_L^*\ g(\tau)\frac{d \tau}{\tau-\tau_1} = \frac{1}{4} g(t) \ , </math>
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| while:
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| :<math>\frac {1}{(2\pi i )^2} \int_L^* g( \tau ) \ d \tau \left( \int_L^* \frac{d \tau_1 } {\left( \tau_1 - t\right) \left( \tau-\tau_1 \right)} \right) = 0 \ . </math>
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| The second form is evaluated using a [[Partial fractions in integration|partial fraction]] expansion and an evaluation using the [[Sokhotski–Plemelj theorem|Sokhotski–Plemelj formula]]:<ref name=Cima>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=http://books.google.com/books?id=1sVLg512ffIC&pg=PA56&dq=%22Plemelj+formula%22&lr=&as_brr=0&sig=Li5rr1v7p1elwQGgf7m7m_dpZHE
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| |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=http://books.google.com/books?id=R3BIOfKssQ4C&pg=PA115&dq=%22Plemelj+formula%22&lr=&as_brr=0&sig=n5qeOMJvwJm8ERxgO3A00VULPs0#PPA101,M1
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| |isbn=0-387-98700-2 |year=1999 |publisher=Springer |edition=2nd Edition }}</ref>
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| :<math>\int_L^*\frac{d \tau_1}{\tau_1-t} = \int_L^* \frac {d\tau_1}{\tau_1-t} = \pi\ i \ . </math>
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| The notation <math>\int_L^*</math> indicates a [[Cauchy principal value]]. See Kanwal.<ref name=Kanwal/>
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| ==Basic theorems==
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| A good discussion of the basis for reversing the order of integration is found in the book ''Fourier Analysis'' by T.W. Körner.<ref name=Körner>{{cite book |title=Fourier Analysis |author=Thomas William Körner |page=Chapters 47 & 48 |url=http://books.google.com/books?id=DZTDtXs4OQAC&printsec=frontcover&dq=Fourier+analysis+subject:%22Fourier+analysis%22&lr=&as_brr=0&sig=KMLyUTINg93Htbfi3vy7GYoSBFw#PPA226,M1
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| |isbn=0-521-38991-7 |publisher=Cambridge University Press |year=1988 }}</ref> 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:
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| :<math>\int_1^{\infty} \frac {x^2-y^2}{\left(x^2+y^2\right)^2}\ dy = \left[\frac{y}{x^2+y^2}\right]_1^{\infty} = -\frac{1}{1+x^2} \ \left[x \ge 1 \right]\ .</math>
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| :::<math>\int_1^{\infty} \left( \int_1^{\infty}\frac {x^2-y^2}{\left(x^2+y^2\right)^2}\ dy \right)\ dx = -\frac{\pi}{4} \ .</math>
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| :::<math>\int_1^{\infty} \left( \int_1^{\infty}\frac {x^2-y^2}{\left(x^2+y^2\right)^2}\ dx \right)\ dy = \frac{\pi}{4} \ .</math>
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| Two basic theorems governing admissibility of the interchange are quoted below from Chaudhry and Zubair:<ref name=Chaudry>{{cite book |title=On a Class of Incomplete Gamma Functions with Applications |author=M. Aslam Chaudhry & Syed M. Zubair |page=Appendix C |url=http://books.google.com/books?id=Edf4KrG_vlYC&pg=PA458&dq=%22order+of+integration%22&lr=&as_brr=0&sig=5XNpVp-EoUsEXruZoX7jI0BakdY |isbn=1-58488-143-7 |publisher=CRC Press |year=2001}}</ref>
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| <div style="border:2px solid #000; background:#FFF; margin:4em; padding:2em;text-indent:3em;">
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| {{Bigbold|Theorem I}}{{Space|5}} Let ''f''(''x'', ''y'') be a continuous function of constant sign defined for ''a ≤ x < ∞'', ''c ≤ y < ∞'', and let the integrals
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| {{Center|<math>J(y):= \int_a^\infty dx \ f(x,\ y)</math>{{space|10}} and {{space|10}}<math>J^*(x) = \int_c^\infty dy\ f(x, \ y)</math>}} regarded as functions of the corresponding parameter be, respectively, continuous for ''c ≤ y < ∞'', ''a ≤ x < ∞''. Then if at least one of the iterated integrals
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| {{Center|<math>\int_c^\infty dy \ \left(\int_a^\infty dx\ f(x,\ y) \right )</math>{{space|10}} and {{space|10}}<math>\int_a^\infty dx \ \left(\int_c^\infty dy\ f(x,\ y) \right )</math>}} converges, the other integral also converges and their values coincide.
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| </div>
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| <div style="border:2px solid #000; background:#FFF; margin:4em; padding:2em;text-indent:3em;">
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| {{Bigbold|Theorem II}}{{Space|5}} Let ''f''(''x'', ''y'') be continuous for ''a ≤ x < ∞'', ''c ≤ y < ∞'', and let the integrals
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| {{Center|<math>J(y):= \int_a^\infty dx \ f(x,\ y)</math>{{space|10}} and {{space|10}}<math>J^*(x) = \int_c^\infty dy\ f(x, \ y)</math>}} 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
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| {{Center|<math>\int_c^\infty dy \ \left(\int_a^\infty dx\ |f(x,\ y)| \right )</math>{{space|10}} and {{space|10}}<math>\int_a^\infty dx \ \left(\int_c^\infty dy\ |f(x,\ y)| \right )</math>}} converges, the iterated integrals
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| {{Center|<math>\int_c^\infty dy \ \left(\int_a^{\infty}dx\ f(x,\ y) \right )</math>{{space|10}} and {{space|10}}<math>\int_a^\infty dx \ \left(\int_c^\infty dy\ f(x,\ y) \right )</math>}} also converge and their values are equal.
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| </div>
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| The most important theorem for the applications is quoted from Protter and Morrey:<ref name=Protter>{{cite book |title=Intermediate Calculus |author=[[Murray H. Protter]] & [[Charles B. Morrey, Jr.]] |page=307 |url=http://books.google.com/books?id=AXw4a2_vzt4C&pg=PA295&dq=%22multiple+integration%22&lr=&as_brr=0&sig=okgH1zN0xLfjhKT-xbf02fXduV4#PPA307,M1 |isbn=0-387-96058-9 |publisher=Springer |year=1985}}</ref>
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| <blockquote style="border:2px solid #000; background:#FFF; margin:4em; padding:2em;text-indent:3em;">
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| Suppose ''F'' is a region given by <math>F=\left\{(x,\ y):a \le x \le b, p(x) \le y \le q(x) \right\} \,</math>  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
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| {{Center|<math> \iint_F f(x,y) dA = \int_a^b\ \int_{p(x)}^{q(x)} f(x,\ y)\,dy\ dx \ .</math>}} The corresponding result holds if the closed region ''F'' has the representation <math>F=\left\{(x,\ y):c\le y \le d,\ r(y) \le x \le s(y)\right\}</math>  where ''r''(''y'') ≤ ''s''(''y'') for ''c ≤ y ≤ d''.  In such a case,
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| : <math> \iint_F f(x,\ y) dA = \int_c^d \ \int_{r(y)}^{s(y)} f(x,\ y)\, dx\ dy \ . </math>
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| <div style="text-indent:0em;">In other words, both iterated integrals, when computable, are equal to the double integral and therefore equal to each other.</div>
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| </blockquote >
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| ==See also==
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| *[[Fubini's theorem]]
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| ==References and notes==
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| <div style="-moz-column-count:2; column-count:2;">
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| <references/>
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| </div>
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| ==External links==
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| *[http://tutorial.math.lamar.edu/Classes/CalcIII/DIGeneralRegion.aspx Paul's Online Math Notes: Calculus III]
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| *[http://www.math.oregonstate.edu/home/programs/undergrad/CalculusQuestStudyGuides/vcalc/255doub/255doub.html] Good 3D images showing the computation of "Double Integrals" using iterated integrals in the Department of Mathematics at Oregon State University.
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| *[http://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)
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| *[http://www.math.umn.edu/~nykamp/m2374/readings/doubleintchange/ Duane Nykamp's University of Minnesota website]
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| *[http://books.google.com/books?id=Q0Pvv4vWOlQC&pg=PA663&dq=%22order+of+integration%22&lr=&as_brr=0&sig=mpTlgi1ymBFa5AesR3AtwiPHKR8] A general introduction.
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| [[Category:Integral calculus]]
| |
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