- #1
aldrinkleys
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Did Fubini's Theorem fail here?
Fubini's Theorem: Did It Fail?
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In summary, Fubini's Theorem is a mathematical theorem that allows for the interchange of integration order in double and repeated integrals. It is commonly used in mathematical analysis to simplify complex integrals and has applications in various fields of mathematics, economics, engineering, and computer science. However, it may fail in certain cases, and alternative methods can be used to evaluate the integral.
Did Fubini's Theorem fail here?
- #2
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Fubini's theorem didn't fail here, because the conditions are not satisfied.
Roughly, Fubini's theorem (in our case) states that if [tex]f:[0,1]^2\rightarrow [-\infty,+\infty][/tex] is a continuous function, such that one of following conditions holds:
1) [tex]f(x,y)\geq 0[/tex] forall x,y in [0,1]
2) [tex]\iint_{[0,1]^2}{|f(x,y)|dxdy}<+\infty[/tex]
then
[tex]\iint_{[0,1]^2}{f(x,y)dxdy}=\int_0^1{\int_0^1{f(x,y)dy}dx}=\int_0^1{\int_0^1{f(x,y)dx}dy}[/tex]
The problem is that neither (1) nor (2) are satisfied in this case.
Roughly, Fubini's theorem (in our case) states that if [tex]f:[0,1]^2\rightarrow [-\infty,+\infty][/tex] is a continuous function, such that one of following conditions holds:
1) [tex]f(x,y)\geq 0[/tex] forall x,y in [0,1]
2) [tex]\iint_{[0,1]^2}{|f(x,y)|dxdy}<+\infty[/tex]
then
[tex]\iint_{[0,1]^2}{f(x,y)dxdy}=\int_0^1{\int_0^1{f(x,y)dy}dx}=\int_0^1{\int_0^1{f(x,y)dx}dy}[/tex]
The problem is that neither (1) nor (2) are satisfied in this case.
- #3
aldrinkleys
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Thank you!
What is Fubini's Theorem?
Fubini's Theorem is a mathematical theorem that states that the order of integration in a double integral and a repeated integral can be interchanged, as long as the resulting integral is convergent.
How is Fubini's Theorem used in mathematical analysis?
Fubini's Theorem is commonly used in mathematical analysis to simplify the evaluation of integrals, particularly in cases where the integrand is difficult to integrate directly. It allows for the breaking down of a complex integral into simpler, one-dimensional integrals that can be computed more easily.
What are some common applications of Fubini's Theorem?
Fubini's Theorem has many applications in various fields of mathematics, including calculus, differential equations, probability, and physics. It is also commonly used in economics, engineering, and computer science.
Has Fubini's Theorem ever failed?
While Fubini's Theorem is generally considered to be a reliable and useful tool in mathematics, there have been cases where it has failed. This can occur when the integrand is not continuous or when the integral is not absolutely convergent.
What can be done if Fubini's Theorem fails?
If Fubini's Theorem fails, alternative methods such as change of variables or integration by parts can be used to evaluate the integral. Additionally, in some cases, the integral may be redefined using a different order of integration to make it convergent.
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