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I have two questions:

How does one apply the Fubini theorem?

and

Can it be applied to a trapezoid?

Thank you.

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3  
You'll need to provide way more context to get a (useful) answer. Apply the theorem of fubini to what? And what do you mean by a trapeze here? –  fgp May 25 '13 at 22:11
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1. Exchange the integrals. 2. Yes. –  Berci May 25 '13 at 22:21
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Don't think so, @MichaelHardy : this really looks like not a real question, with so much context, explanation, ideas, background lacking. I didn't vote to close, but I can see I would easily. –  DonAntonio May 25 '13 at 22:56
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The question is: How is judiciously chosing the order of integration used to evaluate integrals, and does Fubini's theorem justify changes in the order of integration when the domain of the double integral is a trapezoid rather than the Cartesian product of two sets? Is it unclear that that's what's being asked? –  Michael Hardy May 26 '13 at 0:10
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I've started a discussion of this question on meta: meta.math.stackexchange.com/questions/9717/… –  Michael Hardy May 26 '13 at 16:39

1 Answer 1

up vote 14 down vote accepted

Fubini's theorem says a double integral (where the integral of the absolute value is finite) is equal to either of two iterated integrals. One concrete case: $$ \iint\limits_{[a,b]\times[c,d]} f(x,y)\,d(x,y) = \int_a^b\left(\int_c^d f(x,y) \, dy \right) \, dx = \int_c^d \left( \int_a^b f(x,y)\, dx \right) \, dy, $$ provided that $$ \iint\limits_{[a,b]\times[c,d]} |f(x,y)|\,d(x,y) <\infty. $$

One way to use this theorem is to exploit the fact that one of the two iterated integrals may be readily evaluated.

As for trapezoids, here is an example: $$T=\{(x,y) : 1\le y\le2,\ 0\le x\le y \}\tag{1}$$ Suppose one wants $$ \iint\limits_T e^{y^2} \,d(x,y). $$ Applying $(1)$, this becomes $$ \int_1^2 \left( \int_0^y e^{y^2} \, dx \right)\,dy. $$ This is easily evaluated, whereas if one had integrated first with respect to $y$ and afterward with respect to $x$, one would face the intractable integral $\int e^{y^2}\, dy$.

Does Fubini's theorem justify this? The answer is "yes" because one can view the integral as $$ \iint\limits_{[0,2]\times[1,2]} f(x,y) \, d(x,y) $$ where $$ f(x,y)=\begin{cases} e^{y^2} & \text{if }x\le y, \\[10pt] 0 & \text{otherwise}. \end{cases} $$

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+1. Up to multiplication by the characteristic function like you did in the end, trapezoid or not, one can always apply Fubini to an integrable function. –  1015 May 27 '13 at 3:23

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