You should study the double integrals corresponding theory. It is almost useless to select parts of a theory with the only aim of solving a specific problem.
Using the substitution the corresponding Jacobian is so,
on what basis you change the area from R1 to R2
in the integral
??
and in the end why you go back to x y coordinates withoud multiplying by the yacobian
?
and you didnt use the riven R
its between y^2=x and x^2=y