Let's tackle the problems one at a time.

Problem G1 involves a region $R$ in the first quadrant, enclosed by the curves $y = x^2$, $8y = x^2$, $x = y^2$, and $8x = y^2$. To solve the integral $\iint_R \frac{x}{y} \, dA$, we are advised to use a change of variables technique. Analyzing the curves suggests a coordinate transformation based on symmetry or scaling factors.

Problem 2 describes a region $R$ in the $xy$-plane enclosed by an ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$. The problem introduces a transformation $x = a u \cos v$ and $y = b u \sin v$ with the ranges $0 \leq u \leq 1$ and $0 \leq v \leq 2\pi$. Your task here is to verify the Jacobian of the transformation, showing that it equals $ab u$.

Would you like step-by-step solutions for each part?

Here are five questions for further exploration:

1. How would you select a suitable variable transformation in Problem G1 if no hints were provided?
2. Why is the polar coordinate transformation not directly applicable for an ellipse in Problem 2?
3. What does the Jacobian represent in the context of area transformations?
4. How does the change of variables affect the bounds of integration?
5. Can you generalize this approach for any ellipse with different semi-major and semi-minor axes?

Tip: When working with transformations, always calculate the Jacobian determinant to adjust for area distortion in the new coordinates.