Confused about Applying Fubini's Theorem to a Double Integral?

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In summary, the conversation discusses a problem with applying Fubini's theorem to a specific integral. The answer is given as \int_0^a \int_0^t \frac{1}{t}|f(t)|dxdt, but the person is confused about why the second integration domain is set up this way. It is explained that the region is bounded by the curves t=a, x=0, and x=t, and that for a fixed t, x ranges from 0 to t. Once this realization is made, the solution becomes clear.
  • #1
Bernoulli
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Hi, I usually don't have any problems with Fubini's theorem, but there is something I just can't figure out. Let f be integrable, and a some positive constant. How do i apply the theorem to this integral:
[tex] \int_0^a\int_x^a \frac{1}{t}|f(t)|dtdx [/tex]
Really; I know the answer is
[tex]\int_0^a \int_0^t \frac{1}{t}|f(t)|dxdt[/tex]
but I just don't get it. To me this is not obvious (should it be?). Can someone explain this to me?
 
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  • #2
I guess I have made it more difficult then it really is. I just want to know why the second integration domain turns out like that. For simplicity put [tex]\frac{1}{t}|f(t)| = t[/tex] for example (and [tex]t\in (0,a)[/tex]). Then the integration area becomes
[tex]\int_0^a\int_x^a t dtdx = \int_?^?\int_?^?t dxdt[/tex]
Why?
 
  • #3
Draw a picture in the x-t plane. Your region is bounded by the curves t=a, x=0 and x=t. For a fixed t, x ranges from 0 to t. t itself can range from 0 to a.
 
  • #4
Oh, man... I must be tired :) The original problem was not posted like this. I did not realize that the 2D integration area was infact the upper triangle of the square [0,1]^2. I thought I was dealing with the hole square... Stupid me :)

Then of course it is easy.
Thanks
 

FAQ: Confused about Applying Fubini's Theorem to a Double Integral?

What is Fubini's theorem?

Fubini's theorem is a mathematical theorem that states that the order of integration in a double or triple integral can be changed without affecting the result, as long as the function being integrated is continuous.

Why is Fubini's theorem important?

Fubini's theorem is important because it allows for easier calculations and simplification of integrals in higher dimensions. It also provides a way to solve otherwise difficult or impossible integrals.

What are the conditions for using Fubini's theorem?

The conditions for using Fubini's theorem are that the function being integrated must be continuous, the domain of the function must be a rectangle or a bounded region, and the integral must be over a finite region.

What are the applications of Fubini's theorem?

Fubini's theorem has many applications in mathematics, physics, and engineering. It is commonly used in solving problems involving multiple variables, such as calculating volumes and areas, finding probabilities, and solving differential equations.

Are there any limitations to Fubini's theorem?

Yes, there are some limitations to Fubini's theorem. It cannot be used for non-rectangular or unbounded regions, and it may not work for certain functions that are not continuous. It is also important to check for convergence when using Fubini's theorem, as it may not hold for all cases.

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