关于二重积分积分限交换通用形式正确性的技术问询

Hey folks, I'm diving into double integrals and trying to wrap my head around switching the order of integration properly. Let me start by laying out the standard setup I'm working with:

The double integral over the region:
$$
R = \left{ \left( x,: y \right) : a \leqslant x \leqslant b,: g\left( x \right) \leqslant y \leqslant h\left( x \right) \right}
$$
is written as
$$
\iint_R f\left( x,: y \right) \mathrm{d}A = \int_a^b \left[ \int_{g\left( x \right)}^{h\left( x \right)} f\left( x, : y \right) \mathrm{d}y \right] \mathrm{d}x.
$$

From Fubini's theorem, I know that if the function $f$ is integrable, we can reverse the order of integration and get the same result.

Here's where I'm unsure: Assuming $f$ is integrable, and that both $g$ and $h$ are invertible on the interval $a \leqslant x \leqslant b$, is it generally correct to say that the reversed double integral takes the form:
$$
\iint_R f\left( x, : y \right) \mathrm{d}A = \int_{g\lef...
$$

I realize the expression got cut off, but my core question is whether using the inverse functions of $g$ and $h$ directly gives the correct limits for the reversed integral. Has anyone run into this scenario before, or can clarify if this approach holds up in general?

备注：内容来源于stack exchange，提问作者LightninBolt74