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.Theorem I — Let f(x, y) be a continuous function of constant sign defined for a ≤ x < ∞, c ≤ y < ∞, and let the integralsTheorem II — Let f(x, y) be continuous for a ≤ x < ∞, c ≤ y < ∞, and let the integralsTheorem — Suppose F is a region given by F = { ( x ,   y ) : a ≤ x ≤ b , p ( x ) ≤ y ≤ q ( x ) } {displaystyle F=left{(x, y):aleq xleq b,p(x)leq yleq q(x) ight},}   where p and q are continuous and p(x) ≤ q(x) for a ≤ x ≤ b. Suppose that f(x, y) is continuous on F. Then 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.