Order of Integration

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A Order of Integration transforms iterated integrals of functions into other.



References

2016

Integration of order d: A time series is integrated of order d if
[math]\displaystyle{ (1-L)^d X_t \ }[/math]
is a stationary process, where [math]\displaystyle{ L }[/math] is the lag operator and [math]\displaystyle{ 1-L }[/math] is the first difference, i.e.
[math]\displaystyle{ (1-L) X_t = X_t - X_{t-1} = \Delta X. }[/math]
In other words, a process is integrated to order d if taking repeated differences d times yields a stationary process.

2016

Problem statement: The problem for examination is evaluation of an integral of the form
[math]\displaystyle{ \iint_D \ f(x,y ) \ dx \,dy , }[/math]
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.