I need to find non-injective parameterization to some domain such that : $$\int_D f(x,y) dxdy \ne \int_E (f\circ \phi)(u,v)J(u,v) dudv$$
The example I found turns out to be wrong. (The integral is the same when integrating using change of variables and direct computation).
My try :
The domain is $D=\{ 1\le xy \le 2 , 1 \le \frac {x}{y} \le 2\}$ and I want to calculate $\int_D 1dxdy$
The try is wrong, Thanks to @user1027216 comment.
I searched for a counter-example for hours in the internet and couldn't find one.
Any help would be appreciated.
Thank you in advance.