Fubini's Theorem states that for the double integral \(\int_c^d \int_a^b f(x,y)dxdy\), the order of integration can be exchanged if \(\int_c^d \int_a^b |f(x,y)|dxdy < \infty\). This discussion confirms that the condition of finiteness for the integral of the absolute value, \(\int_c^d \int_a^b |f(x,y)|dxdy\), must hold for both orders of integration, but if it is finite for one order, it is also finite for the other. Therefore, the less-than-infinity property is consistent across both iterations of the integral.
PREREQUISITESMathematicians, students of calculus, and anyone studying advanced integration techniques will benefit from this discussion on Fubini's Theorem and its implications for double integrals.
Yes (for |f(x,y)|, the order doesn't matter).Poopsilon said:So given [itex]\int_c^d \int_a^b f(x,y)dxdy[/itex], we can exchange the order of the integrals provided that [itex]\int_c^d \int_a^b |f(x,y)|dxdy < \infty[/itex]. Does this less-than-infinity property have to hold for both orders of iteration i.e. for dxdy and dydx? Or can it be proven that if it's finite for one order than it's finite for the other?