The sequence in which variables are integrated in an iterated integral. Reversing the order changes the slicing of the domain of integration, often requiring more than one integral. Fubini's theorem permits interchange for absolutely integrable functions, while Tonelli's theorem permits nonnegative measurable integrands even when the integral is infinite. Conditional convergence requires additional care; geometric reversal alone is not a general justification for interchanging improper integrals.
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The order of integration refers to the number of times a function has been integrated. In calculus, the process of integration can be performed multiple times, and each layer of integration adds to the "order." Here’s a brief breakdown of the concept: 1. **First Order Integration**: This is the process of integrating a function once.