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Topics in Calculus
Differential calculus
	Derivative; Change of variables; Implicit differentiation; Taylor's theorem; Related rates; Identities; Rules:; Power rule, Product rule, Quotient rule, Chain rule
Integral calculus
	Integral; Lists of integrals; Improper integrals; Integration by:; parts, disks, cylindrical; shells, substitution,; trigonometric substitution,; partial fractions, changing order
Vector calculus
	Gradient; Divergence; Curl; Laplacian; Gradient theorem; Green's theorem; Stokes' theorem; Divergence theorem
Multivariable calculus
	Matrix calculus; Partial derivative; Multiple integral; Line integral; Surface integral; Volume integral; Jacobian

For the summary statistic in time series, see Order of integration.

In calculus, interchange of the order of integration is a methodology that transforms multiple integrations 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.

[edit] Problem statement

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

$\iint_D \ f(x,y ) \ dx \,dy ,$

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. A major headache with this interchange is determining the change in description of the domain D.

The method also is applicable to multiple integrals.^[1]^[2]

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.

[edit] Relation to integration by parts

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.

Consider the double integral (see the Multiple_integral#Formulae_of_reduction article section for explanation of the following notation):

$\int_a^z \ dx\ \int_a^x \ h(y) dy \$

In the order written above, the 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) as shown in the left panel of Figure 1, which is inconvenient especially when 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:

$\int_a^z \ dx\ \int_a^x \ h(y) \ dy \ = \int_a^z \ h(y)\ dy \ \ \int_y^z \ dx = \int_a^z \ \left(z-y\right) h(y)\, dy \ .$

This result can be seen to be an example of the formula for integration by parts, as stated below:^[3]

$\int_a^z f(x) g'(x)\, dx = \left[ f(x) g(x) \right]_a^z - \int_a^z f'(x) g(x)\, dx\!$

Substitute:

$g (x) = \int_a^x \ h(y)\, dy \text{ and }f(x) = z-x \ .$

However, compared to using the formula for integration by parts, exchange of the order of integration has the merit that it generates the function f in a natural manner.

[edit] General examples

More complex examples of changing the order of integration can be found at Ron Miech's UCLA Calculus Problems (see Problems 33, 35, 37, 39, 41 & 43) and Duane Nykamp's University of Minnesota website. For a general introduction, see Murthy and Srinivas^[4], Widder^[5], or Johnson.^[6]

[edit] Principal-value integrals

For application to principal-value integrals, see Whittaker and Watson,^[7] , Gakhov,^[8] Lu^[9], or Zwillinger.^[10] See also the discussion of the Poincaré-Bertrand transformation in Obolashvili.^[11] An example where the order of integration cannot be exchanged is given by Kanwal:^[12]

$\frac {1}{(2\pi i )^2} \int_L^* \frac{d{\tau}_1}{{\tau}_1 - t}\ \int_L^*\ g(\tau)\frac{d \tau}{\tau-\tau_1} = \frac{1}{4} g(t) \ ,$

while:

$\frac {1}{(2\pi i )^2} \int_L^* g( \tau ) \ d \tau \left( \int_L^* \frac{d \tau_1 } {\left( \tau_1 - t\right) \left( \tau-\tau_1 \right)} \right) = 0 \ .$

The second form is evaluated using a partial fraction expansion and an evaluation using the Sokhotski-Plemelj formula:^[13]

$\int_L^*\frac{d \tau_1}{\tau_1-t} = \int_L^* \frac {d\tau_1}{\tau_1-t} = \pi\ i \ .$

The notation $\int_L^*$ indicates a Cauchy principal value. See Kanwal.^[12]

[edit] Basic theorems

A good discussion of the basis for reversing the order of integration is found in Körner.^[14] 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:

$\int_1^{\infty} \frac {x^2-y^2}{\left(x^2+y^2\right)^2}\ dy = \left[\frac{y}{x^2+y^2}\right]_1^{\infty} = -\frac{1}{1+x^2} \ \left[x \ge 1 \right]\ .$

$\int_1^{\infty} \left( \int_1^{\infty}\frac {x^2-y^2}{\left(x^2+y^2\right)^2}\ dy \right)\ dx = -\frac{\pi}{4} \ .$

$\int_1^{\infty} \left( \int_1^{\infty}\frac {x^2-y^2}{\left(x^2+y^2\right)^2}\ dx \right)\ dy = \frac{\pi}{4} \ .$

Two basic theorems governing admissibility of the interchange are quoted below from Chaudhry and Zubair:^[15]

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

$J(y):= \int_a^\infty dx \ f(x,\ y)$ and $J^*(x) = \int_c^\infty dy\ f(x, \ y)$

regarded as functions of the corresponding parameter be, respectively, continuous for c ≤ y < ∞, a ≤ x < ∞. Then if at least one of the iterated integrals

$\int_c^\infty dy \ \left(\int_a^\infty dx\ f(x,\ y) \right )$ and $\int_a^\infty dx \ \left(\int_c^\infty dy\ f(x,\ y) \right )$

converges, the other integral also converges and their values coincide.

Theorem II Let f(x, y) be continuous for a ≤ x < ∞, c ≤ y < ∞, and let the integrals

$J(y):= \int_a^\infty dx \ f(x,\ y)$ and $J^*(x) = \int_c^\infty dy\ f(x, \ y)$

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

$\int_c^\infty dy \ \left(\int_a^\infty dx\ |f(x,\ y)| \right )$ and $\int_a^\infty dx \ \left(\int_c^\infty dy\ |f(x,\ y)| \right )$

converges, the iterated integrals

$\int_c^\infty dy \ \left(\int_a^{\infty}dx\ f(x,\ y) \right )$ and $\int_a^\infty dx \ \left(\int_c^\infty dy\ f(x,\ y) \right )$

also converge and their values are equal.

The most important theorem for the applications is quoted from Protter and Morrey:^[16]

Suppose F is a region given by $F=\left\{(x,\ y):a \le x \le b, p(x) \le y \le q(x) \right\} \,$ 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

$\iint_F f(x,y) dA = \int_a^b\ \int_{p(x)}^{q(x)} f(x,\ y)\,dy\ dx \ .$

The corresponding result holds if the closed region F has the representation $F=\left\{(x,\ y):c\le y \le d,\ r(y) \le x \le s(y)\right\}$ where r(y) ≤ s(y) for c ≤ y ≤ d. In such a case,

$\iint_F f(x,\ y) dA = \int_c^d \ \int_{r(y)}^{s(y)} f(x,\ y)\, dx\ dy \ .$

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

[edit] See also

Fubini's theorem

[edit] References and notes

^ Seán Dineen (2001). Multivariate Calculus and Geometry. Springer. p. 162. ISBN 185233472X. http://books.google.com/books?id=1YNX3YAf1vMC&pg=PA165&dq=%22order+of+integration%22&lr=&as_brr=0&sig=hF51z9mPjvfq_lisOmHwuLUrM0Q#PPA162,M1.
^ Richard Courant & Fritz John (2000). Introduction to Calculus and Analysis: Vol. II/1, II/2. Classics in mathematics. Springer. p. 897. ISBN 3540665692. 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.
^ The prime notation " ′ " denotes a derivative.
^ Ch V Ramana Murthy & NC Srinivas (2001). Applied Mathematics. New Delhi: S.Chand & Company Ltd.. p. 663. ISBN 8121920825. http://books.google.com/books?id=Q0Pvv4vWOlQC&pg=PA663&dq=%22order+of+integration%22&lr=&as_brr=0&sig=mpTlgi1ymBFa5AesR3AtwiPHKR8.
^ David Vernon Widder (1989). Advanced Calculus: Second Edition. Dover Publications. p. 195. ISBN 0486661032. http://books.google.com/books?id=Cl7MhMC3E7AC&pg=PA195&dq=%22order+of+integration%22&lr=&as_brr=0&sig=M5TBOJF5clQ4B4mr4PqEN7Elwq0.
^ William Woolsey Johnson (1898). An Elementary Treatise on the Integral Calculus: Founded on the method of rates or fluxions. Wiley. p. 158ff. http://books.google.com/books?id=rC0AAAAAYAAJ&pg=PA158&dq=%22order+of+integration%22&lr=&as_brr=0#PPA158,M1.
^ Edmund Taylor Whittaker & George Neville Watson (1927). 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 (4th ed., repr ed.). Cambridge University Press. p. §4.51, p. 75. ISBN 0521588073. http://books.google.com/books?id=ULVdGZmi9VcC&pg=PA75&dq=principal+value+%22order+of++integration%22&lr=&as_brr=0&sig=bx7JymRGBqKQ-K9KHRwRJk8XoYI.
^ F. D. Gakhov (1990). Boundary Value Problems. Courier Dover Publications. p. 46. ISBN 0486662756. http://books.google.com/books?id=9G7sfwTDv8QC&pg=PA46&dq=%22order+of+integration%22&lr=&as_brr=0&sig=jw8C_74EzuwTmHXOAPRX9KrNi5g.
^ Jian-Ke Lu (1993). Boundary Value Problems for Analytic Functions. Singapore: World Scientific. p. 44. ISBN 9810210205. 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.
^ Daniel Zwillinger (1992). Handbook of integration. AK Peters Ltd.. p. 61. ISBN 0867202939. http://books.google.com/books?id=DQd4wfV7fo0C&pg=PA61&dq=%22order+of+integration%22&lr=&as_brr=0&sig=pev_kroO46LEqZcxY1qJA01qv9U.
^ Elena Irodionovna Obolashvili (2003). Higher order partial differential equations in Clifford analysis: effective solutions to problems. Birkhäuser. p. 101. ISBN 0817642862. http://books.google.com/books?id=HmvmB6NCyEAC&pg=PA101&dq=principal+value+%22order+of++integration%22&lr=&as_brr=0&sig=1ZcvrKq8hD6Zf-6PL2Bo6H0KEp0.
^ ^a ^b Ram P. Kanwal (1996). Linear Integral Equations: theory and technique (2nd Edition ed.). Boston: Birkhäuser. p. 194. ISBN 0817639403. 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.
^ For a discussion of the Sokhotski-Plemelj formula see, for example, Joseph A. Cima, Alec L. Matheson & William T. Ross (2006). The Cauchy Transform. American Mathematical Society. p. 56. ISBN 0821838717. http://books.google.com/books?id=1sVLg512ffIC&pg=PA56&dq=%22Plemelj+formula%22&lr=&as_brr=0&sig=Li5rr1v7p1elwQGgf7m7m_dpZHE. or Rainer Kress (1999). Linear integral equations (2nd Edition ed.). Springer. p. Theorem 7.6, p. 101. ISBN 0387987002. http://books.google.com/books?id=R3BIOfKssQ4C&pg=PA115&dq=%22Plemelj+formula%22&lr=&as_brr=0&sig=n5qeOMJvwJm8ERxgO3A00VULPs0#PPA101,M1.
^ Thomas William Körner (1988). Fourier Analysis. Cambridge University Press. p. Chapters 47 & 48. ISBN 0521389917. http://books.google.com/books?id=DZTDtXs4OQAC&printsec=frontcover&dq=Fourier+analysis+subject:%22Fourier+analysis%22&lr=&as_brr=0&sig=KMLyUTINg93Htbfi3vy7GYoSBFw#PPA226,M1.
^ M. Aslam Chaudhry & Syed M. Zubair (2001). On a Class of Incomplete Gamma Functions with Applications. CRC Press. p. Appendix C. ISBN 1584881437. http://books.google.com/books?id=Edf4KrG_vlYC&pg=PA458&dq=%22order+of+integration%22&lr=&as_brr=0&sig=5XNpVp-EoUsEXruZoX7jI0BakdY.
^ Murray H. Protter & Charles B. Morrey, Jr. (1985). Intermediate Calculus. Springer. p. 307. ISBN 0387960589. http://books.google.com/books?id=AXw4a2_vzt4C&pg=PA295&dq=%22multiple+integration%22&lr=&as_brr=0&sig=okgH1zN0xLfjhKT-xbf02fXduV4#PPA307,M1.

[edit] External links

Paul's Online Math Notes: Calculus III

[Dineen-0] Seán Dineen (2001). Multivariate Calculus and Geometry. Springer. p. 162. ISBN 185233472X. http://books.google.com/books?id=1YNX3YAf1vMC&pg=PA165&dq=%22order+of+integration%22&lr=&as_brr=0&sig=hF51z9mPjvfq_lisOmHwuLUrM0Q#PPA162,M1.

[Courant-1] Richard Courant & Fritz John (2000). Introduction to Calculus and Analysis: Vol. II/1, II/2. Classics in mathematics. Springer. p. 897. ISBN 3540665692. 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.

[2] The prime notation " ′ " denotes a derivative.

[Murthy-3] Ch V Ramana Murthy & NC Srinivas (2001). Applied Mathematics. New Delhi: S.Chand & Company Ltd.. p. 663. ISBN 8121920825. http://books.google.com/books?id=Q0Pvv4vWOlQC&pg=PA663&dq=%22order+of+integration%22&lr=&as_brr=0&sig=mpTlgi1ymBFa5AesR3AtwiPHKR8.

[Widder-4] David Vernon Widder (1989). Advanced Calculus: Second Edition. Dover Publications. p. 195. ISBN 0486661032. http://books.google.com/books?id=Cl7MhMC3E7AC&pg=PA195&dq=%22order+of+integration%22&lr=&as_brr=0&sig=M5TBOJF5clQ4B4mr4PqEN7Elwq0.

[Johnson-5] William Woolsey Johnson (1898). An Elementary Treatise on the Integral Calculus: Founded on the method of rates or fluxions. Wiley. p. 158ff. http://books.google.com/books?id=rC0AAAAAYAAJ&pg=PA158&dq=%22order+of+integration%22&lr=&as_brr=0#PPA158,M1.

[Whittaker-6] Edmund Taylor Whittaker & George Neville Watson (1927). 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 (4th ed., repr ed.). Cambridge University Press. p. §4.51, p. 75. ISBN 0521588073. http://books.google.com/books?id=ULVdGZmi9VcC&pg=PA75&dq=principal+value+%22order+of++integration%22&lr=&as_brr=0&sig=bx7JymRGBqKQ-K9KHRwRJk8XoYI.

[Gakhov-7] F. D. Gakhov (1990). Boundary Value Problems. Courier Dover Publications. p. 46. ISBN 0486662756. http://books.google.com/books?id=9G7sfwTDv8QC&pg=PA46&dq=%22order+of+integration%22&lr=&as_brr=0&sig=jw8C_74EzuwTmHXOAPRX9KrNi5g.

[Lu-8] Jian-Ke Lu (1993). Boundary Value Problems for Analytic Functions. Singapore: World Scientific. p. 44. ISBN 9810210205. 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.

[Zwillinger-9] Daniel Zwillinger (1992). Handbook of integration. AK Peters Ltd.. p. 61. ISBN 0867202939. http://books.google.com/books?id=DQd4wfV7fo0C&pg=PA61&dq=%22order+of+integration%22&lr=&as_brr=0&sig=pev_kroO46LEqZcxY1qJA01qv9U.

[Obolashvili-10] Elena Irodionovna Obolashvili (2003). Higher order partial differential equations in Clifford analysis: effective solutions to problems. Birkhäuser. p. 101. ISBN 0817642862. http://books.google.com/books?id=HmvmB6NCyEAC&pg=PA101&dq=principal+value+%22order+of++integration%22&lr=&as_brr=0&sig=1ZcvrKq8hD6Zf-6PL2Bo6H0KEp0.

[Kanwal-11] Ram P. Kanwal (1996). Linear Integral Equations: theory and technique (2nd Edition ed.). Boston: Birkhäuser. p. 194. ISBN 0817639403. 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.

[Cima-12] For a discussion of the Sokhotski-Plemelj formula see, for example, Joseph A. Cima, Alec L. Matheson & William T. Ross (2006). The Cauchy Transform. American Mathematical Society. p. 56. ISBN 0821838717. http://books.google.com/books?id=1sVLg512ffIC&pg=PA56&dq=%22Plemelj+formula%22&lr=&as_brr=0&sig=Li5rr1v7p1elwQGgf7m7m_dpZHE. or Rainer Kress (1999). Linear integral equations (2nd Edition ed.). Springer. p. Theorem 7.6, p. 101. ISBN 0387987002. http://books.google.com/books?id=R3BIOfKssQ4C&pg=PA115&dq=%22Plemelj+formula%22&lr=&as_brr=0&sig=n5qeOMJvwJm8ERxgO3A00VULPs0#PPA101,M1.

[13] Thomas William Körner (1988). Fourier Analysis. Cambridge University Press. p. Chapters 47 & 48. ISBN 0521389917. http://books.google.com/books?id=DZTDtXs4OQAC&printsec=frontcover&dq=Fourier+analysis+subject:%22Fourier+analysis%22&lr=&as_brr=0&sig=KMLyUTINg93Htbfi3vy7GYoSBFw#PPA226,M1.

[Chaudry-14] M. Aslam Chaudhry & Syed M. Zubair (2001). On a Class of Incomplete Gamma Functions with Applications. CRC Press. p. Appendix C. ISBN 1584881437. http://books.google.com/books?id=Edf4KrG_vlYC&pg=PA458&dq=%22order+of+integration%22&lr=&as_brr=0&sig=5XNpVp-EoUsEXruZoX7jI0BakdY.

[Protter-15] Murray H. Protter & Charles B. Morrey, Jr. (1985). Intermediate Calculus. Springer. p. 307. ISBN 0387960589. http://books.google.com/books?id=AXw4a2_vzt4C&pg=PA295&dq=%22multiple+integration%22&lr=&as_brr=0&sig=okgH1zN0xLfjhKT-xbf02fXduV4#PPA307,M1.

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