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Definition: Fubini's theorem, sometimes called Tonelli's theorem, establishes a connection between a multiple integral and a repeated one. If f(x,y) is continuous on the rectangular region R:a<=x<=b,c<=y<=d, then the equality int int_R [f(x,y)d(x,y)]=int_(a to b) int_(c to d) [f(x,y)] dydx holds.