I want to prove, that area of the hyperbolic sector is invariant under tranformation of the form:
$f((x,y)) = (sx, \frac{y}{s})$, where $s$ is arbitrary positive number. Let's find the area, it should be equal to integral $A=\int_{a}^{b} \frac{1}{x} dx = \ln{\frac{a}{b}}$. Then $\ln{\frac{sa}{sb}} = \ln{\frac{a}{b}}$. Am I correct? It confuses me that I didn’t use the condition for the second coordinate in any way here.
