Order Of Integration (calculus)

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 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.

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

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 ''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. 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:

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

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

:\int_a^z f(x) g'(x)\, dx = \left f(x) g(x) \righta^z - \int_a^z f'(x) g(x)\, dx

Substitute:

:$f (x) = \int_a^x h(y)\, dy ~\text~ g'(x) = 1 .$

Which gives the result.

Principal-value integrals

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

:$\frac \int_L^* \frac\ \int_L^*\ g(\tau)\frac = \frac g(t) \ ,$

while:

:$\frac \int_L^* g( \tau ) \ d \tau \left( \int_L^* \frac \right) = 0 \ .$

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, or

:$\int_L^*\frac = \int_L^* \frac = \pi\ i \ .$

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

Basic theorems

A discussion of the basis for reversing the order of integration is found in the book ''Fourier Analysis'' by T.W. Körner. 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^ \frac \ dy = \left frac\right1^ = -\frac \ \left \ge 1 \right .

:::$\int_1^ \left( \int_1^\frac \ dy \right)\ dx = -\frac \ .$
:::$\int_1^ \left( \int_1^\frac \ dx \right)\ dy = \frac \ .$

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

The most important theorem for the applications is quoted from Protter and Morrey:

See also

* Fubini's theorem

References and notes

External links

Paul's Online Math Notes: Calculus III
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

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Integral calculus