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?
Fubini's Theorem is a mathematical theorem that explains the relationship between double integrals and iterated integrals. It states that if a function is integrable over a rectangle in the xy-plane, then the double integral of the function can be calculated as an iterated integral.
Fubini's Theorem is significant because it allows us to evaluate double integrals using simpler iterated integrals. This makes it easier to calculate the integral of a function over a 2-dimensional region, which has many applications in physics, engineering, and other fields.
Yes, Fubini's Theorem can be extended to functions of more than two variables. It is known as the Tonelli's Theorem and it states that if a function is integrable over a product of n-dimensional rectangles, then the n-fold integral of the function can be calculated as an iterated integral.
For Fubini's Theorem to hold, the function must be integrable over the rectangular region of integration. Additionally, one of the following conditions must be satisfied: the function must be continuous, the function must be piecewise continuous, or the function must be bounded and have a finite number of discontinuities.
Yes, Fubini's Theorem can be applied to non-rectangular regions by using a change of variables. This allows us to transform the non-rectangular region into a rectangular one, making it possible to apply Fubini's Theorem. However, the conditions for the theorem to hold still apply.