Change of coordinate system when integrating

Suppose we have two coordinate systems: (representing the usual Cartesian coordinates, for example) and (representing cylindrical polars, say). I think it's true that:

,

but I can't rigorously prove this. (It feels right based on the chain rule but I can't write a formal proof, and haven't found anything online.) Can anyone help please?

On a similar note, for functions is it true that

?

I'm less sure that this second formula holds in general.

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Re: Change of coordinate system when integrating