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I was looking over the triple integral below:

And I was wondering, when exactly are we allowed to break up a triple integral into the product of its components?

Robert
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    when all six integral endpoints are constants and the integrand is the product of one-variable functions – Will Jagy Apr 05 '15 at 19:50

2 Answers2

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With Fubini-Tonelli Theorem you can prove that $$ \iint_{X \times Y} f(x)g(y)\;\mathrm dx\;\mathrm dy = \int_X f(x)\;\mathrm dx\int_Y\;g(y)\mathrm dy $$ Provided that $X,Y$ are independent of $y$ and $x$.

Hence a integral can be broken down into factors when the ranges of integration are independent.

Henricus V.
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As Will says, you can break up a multiple integral into the product of single integrals only when all the endpoints are constant (you are integrating in a box) and the integrand is a product of those one variable functions. You can also likewise see that you can split up a multiple integral into the product of smaller integrals if those integrals are 'independent' of each other, i.e. no variable from one appears in the others.

Bob Krueger
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