Fubini's theorem

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Fubini's theorem gives the conditions under which a double integral can be computed as an iterated integral, i.e. by integrating in one variable at a time. Intuitively, just as the volume of a loaf of bread is the same whether one sums over standard slices or over long thin slices, the value of a double integral does not depend on the order of integration when the hypotheses of the theorem are satisfied. The theorem is named after Guido Fubini, who proved a general result in 1907; special cases were known earlier through results such as Cavalieri's principle, which was used by Leonhard Euler. More formally, the theorem states that if a function is Lebesgue integrable on a rectangle X × Y {\displaystyle X\times Y} , then one can evaluate the double integral as an iterated integral: ∬ X × Y f ( x...

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