Finding a variable substitution for a double integral I have the following double integral
$$
\iint_D xdxdy
$$
where D is given by the inequalities
$$
x^2+xy+y^2 \le 4, x\ge 0
$$
In the solution given to the problem they apply this variable substitution when solving the integral:
$$
u = \frac{\sqrt{3}}{2}x
$$
$$
v = \frac{x}2 + y
$$
My question is how they arrive at this substitution? Is there a method for finding substitutions like these?
While I understand that this substitution works, there is no explanation for how they arrive at this substitution at all in my solution.

We want to represent x as a function of y for the first constraint. So we complete to square with respect to x:
$$y^2+2(y)(x/2)+x^2/4+\frac 3 4x^2=(y+x/2)^2+\frac 3 4x^2<4$$
You don't have to use any change of variables, you could simply write y in terms of x
