Part of a series about

Calculus




Definitions


Integration by








In calculus, interchange of the order of integration is a methodology that transforms iterated integrals (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.
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 twodimensional 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.^{[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.
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 zaxis onto the xy plane. The sloped line is the curve y = x.
Consider the iterated integral

\int_a^z \, \int_a^x \, h(y) \, dy \, dx \ ,
which we will write using the prefix notation commonly seen in physics:

\int_a^z dx \, \int_a^x \, h(y) \, dy .
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 ydirection (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 yaxis. This forms a three dimensional slice dx wide along the xaxis, 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.^{[3]} 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:

\int_a^z \ dx\ \int_a^x \ h(y) \ dy \ = \int_a^z \ h(y)\ dy \ \ \int_y^z \ dx = \int_a^z \ \left(zy\right) h(y)\, dy \ .
This result can be seen to be an example of the formula for integration by parts, as stated below:^{[4]}

\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:

f (x) = \int_a^x \ h(y)\, dy \text{ and }g'(x) = 1 \ .
Which gives the result.
Principalvalue integrals
For application to principalvalue integrals, see Whittaker and Watson,^{[5]} Gakhov,^{[6]} Lu,^{[7]} or Zwillinger.^{[8]} See also the discussion of the PoincaréBertrand transformation in Obolashvili.^{[9]} An example where the order of integration cannot be exchanged is given by Kanwal:^{[10]}

\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:^{[11]}

\int_L^*\frac{d \tau_1}{\tau_1t} = \int_L^* \frac {d\tau_1}{\tau_1t} = \pi\ i \ .
The notation \int_L^* indicates a Cauchy principal value. See Kanwal.^{[10]}
Basic theorems
A good discussion of the basis for reversing the order of integration is found in the book Fourier Analysis by T.W. Körner.^{[12]} 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^2y^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^2y^2}{\left(x^2+y^2\right)^2}\ dy \right)\ dx = \frac{\pi}{4} \ .

\int_1^{\infty} \left( \int_1^{\infty}\frac {x^2y^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:^{[13]}
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:^{[14]}
Theorem — 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.
See also
References and notes

^

^

^

^ The prime notation " ′ " denotes a derivative.

^

^

^

^

^

^ ^{a} ^{b}

^ For a discussion of the SokhotskiPlemelj formula see, for example, or

^

^

^
External links

Paul's Online Math Notes: Calculus III

Good 3D images showing the computation of "Double Integrals" using iterated integrals, the Department of Mathematics at Oregon State University.

Ron Miech's UCLA Calculus Problems More complex examples of changing the order of integration (see Problems 33, 35, 37, 39, 41 & 43)

Duane Nykamp's University of Minnesota website

A general introduction.
This article was sourced from Creative Commons AttributionShareAlike License; additional terms may apply. World Heritage Encyclopedia content is assembled from numerous content providers, Open Access Publishing, and in compliance with The Fair Access to Science and Technology Research Act (FASTR), Wikimedia Foundation, Inc., Public Library of Science, The Encyclopedia of Life, Open Book Publishers (OBP), PubMed, U.S. National Library of Medicine, National Center for Biotechnology Information, U.S. National Library of Medicine, National Institutes of Health (NIH), U.S. Department of Health & Human Services, and USA.gov, which sources content from all federal, state, local, tribal, and territorial government publication portals (.gov, .mil, .edu). Funding for USA.gov and content contributors is made possible from the U.S. Congress, EGovernment Act of 2002.
Crowd sourced content that is contributed to World Heritage Encyclopedia is peer reviewed and edited by our editorial staff to ensure quality scholarly research articles.
By using this site, you agree to the Terms of Use and Privacy Policy. World Heritage Encyclopedia™ is a registered trademark of the World Public Library Association, a nonprofit organization.