Transformation to uv plane integral, Carnot engine

I'm curious how do you choose the u and v transformations here to be equal to the constants and why is the f(u,v)=1 for the area? I see that the area in xy is difficult to integrate because the sides are curved. is the transformation proven in a different book than Stewart? the problem: the work done by an ideal Carnot engine is equal to the area enclosed by two isotherms and adiabatic curves. xy is a hyperbola

by the way, how long does it take people to do these?

Re: Transformation to uv plane integral, Carnot engine

Re: Transformation to uv plane integral, Carnot engine

it's interesting it doesn't matter if you reverse u=y-x and v=y+x when transforming to the uv plane

is it bad to just map the vertices instead of lines?