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**Using the "Product of the Integrals is integrals of Product" Magic in multivar**

This isn't actually a question on a homework problem, but It is coursework related so I thought it belonged here. I've noticed a lot of magic in Griffiths and other textbooks, where say, in integrating in spherical co-ordinates, you simply do the theta, phi, and r integrals, and them multiply them all together...

but I thought this was not a valid way to do integration... like it makes sense to me, but I can't justify to myself why it would always be true that one could do this... is it maybe only true when the function your integrating doesn't depend on all three variables, or you can get it into a form where the product of these things don't?

For instance, Int (e^-x^2-y^2)dxdy.... across all of R^2 this integral of this is equal to the integral of e^-x^2 dx times from 0-> infinity the integral of e^-y^2 dy multiplied together from 0 to infinity... Why? and why is it legit to do this "pulling out dphi and multiplying by that integral" like Griffiths does in his electrodynamics?

Thanks in advance.