PX275 - B3a - hyperbolic coordinates

consider the thermodynamics of an ideal gas: $p V = n R T$
whilst, it can be integrated over the area, $Ω$ , it would be better to find another region
the problem is simplified with hyperbolic coordinates:

p V = r; \frac{p}{V} = s

J = \frac{\partial (p, V)}{\partial (r, s)}

I_{Ω} = \int_{s_{1}}^{s_{2}} \int_{r_{1}}^{r_{2}} | J | d r d s