Question: I was looking over the triple integral below:

![](https://i.stack.imgur.com/SBX8p.png)

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

Answer: 
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.
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.