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.

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

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.

Consider the iterated integral

which we will write using the prefix notation commonly seen in physics:

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. 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:

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

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