Okay so I know the idea of solving these types of problems, but what I'm concerned about is when we have to use a change of variable formula and when not to... specifically, for this example we have to:
Let $u = \frac x2$ and $w = \frac y3$, so $x = 2u$ and $y = 3w$, plugging this into the jacobian matrix we get 6 as the determinant.
Then we plug in $x = 2u$ into the equation, $x^2$, multiplied by the determinant which is absolute value of $6 \text{ dudw}$.
I though all we needed to do was get the domain in polar coordinates, $(r,\theta)$ and then sub $x = r \cos \theta, y = r \sin \theta$, set up the integral for the equation following $r \text{drd}\theta$ and solve?
Actually you can use elliptic coordinates immediatelly, which are nothing else but a little bit modified polar coordinates. Write $x=2r \cos \theta$ and $y = 3r \cos \theta$ and the Jacobian will be $6r$ and $0 \le r \le 1$ and $0 \le \theta \le 2\pi$
In fact what you did is exactly this, but in an explicit form. The region $D$ is an ellipse. The linear transformation $x=2u$ and $y=3v$ in fact streches it into a circle. Then you can use the polar coordinates to solve the integral. The first way does the two steps at once.