# General formula for reversing double integral bounds

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 expressed as $$\iint_R f\left( x,\: y \right) \mathrm{d}A = \int_a^b \left[ \int_{g\left( x \right)}^{h\left( x \right)} f\left( x, \: y \right) \mathrm{d}y \right] \mathrm{d}x.$$ By Fubini's theorem, any permutation of the order of integration is equivalent if the function $$f$$ is integrable.

Assuming $$f$$ is integrable, if $$g$$ and $$h$$ are invertible on the interval $$a \leqslant x \leqslant b$$, is it correct to say that in general, the reversed double integral has the form: $$\iint_R f\left( x, \: y \right) \mathrm{d}A = \int_{g\left( a \right)}^{h\left( b \right)} \left[ \int_{h^{-1}\left( y \right)}^{g^{-1}\left( y \right)} f\left( x, \: y \right) \mathrm{d}x \right] \mathrm{d}y.$$ Would there be any other restrictions on the functions $$g$$ and $$h$$?

The area of the double integral with the reversed bounds differs from that of the first integral, which can be easily seen from the following figure; compare the two ranges $$(a,b)$$ and $$\left (h^{-1}(g(a)),g^{-1}(h(b))\right).$$