Let's analyze the problem from the image step by step.

1. Calculate the Jacobian

The image defines the change of variables $x = 2u$ and $y = 3v$ to transform the coordinates of the region enclosed by the ellipse:

$\frac{x^2}{4} + \frac{y^2}{9} = 1$

The Jacobian determinant is given by:

$J(u, v) = \left| \frac{\partial(x, y)}{\partial(u, v)} \right|$

Where $x = 2u$ and $y = 3v$. Now let's compute the partial derivatives:

$\frac{\partial x}{\partial u} = 2, \quad \frac{\partial x}{\partial v} = 0$ $\frac{\partial y}{\partial u} = 0, \quad \frac{\partial y}{\partial v} = 3$

So the Jacobian determinant is:

$J(u, v) = \left| \begin{matrix} 2 & 0 \\ 0 & 3 \end{matrix} \right| = 2 \times 3 = 6$

Thus, $J(u, v) = 6$, which matches the provided answer.

2. Rewrite the integral using the change of variables

We are asked to rewrite the given integral over the region $D$ (the upper part of the ellipse) by changing the variables to $u$ and $v$:

$\iint_D y \, dA = \iint_S f(u, v) \, du \, dv$

First, we need to express the function $f(u, v)$ and determine the region $S$.

a) Function $f(u, v)$:

Since $y = 3v$ and the Jacobian is 6, the new function is:

$f(u, v) = y \times \text{Jacobian} = 3v \times 6 = 18v$

Thus, $f(u, v) = 18v$, which is already filled in the provided image.

b) Region $S$:

The region $D$ is the part of the ellipse $\frac{x^2}{4} + \frac{y^2}{9} = 1$ that lies above the $x$-axis.

Using the transformations $x = 2u$ and $y = 3v$, the equation of the ellipse becomes:

$\frac{(2u)^2}{4} + \frac{(3v)^2}{9} = 1 \quad \Rightarrow \quad u^2 + v^2 = 1$

Since we are only considering the part of the ellipse above the $x$-axis (i.e., $y \geq 0$), we require $v \geq 0$.

Thus, the region $S$ is described by $u^2 + v^2 \leq 1$ and $v \geq 0$. This corresponds to option c:

$S = \{(u, v) \mid u^2 + v^2 \leq 1, v \geq 0 \}$

Summary:

1. The Jacobian is $J(u, v) = 6$.
2. The function $f(u, v) = 18v$ and the region $S$ is given by option c: $\{(u, v) \mid u^2 + v^2 \leq 1, v \geq 0 \}$.

Would you like any further clarifications or more details on this? Here are some related questions:

1. How do you compute the Jacobian in more complex variable transformations?
2. What is the significance of the Jacobian in coordinate transformations?
3. Can you extend this method to non-elliptical regions?
4. How would the integral change if the region was the entire ellipse?
5. How does the orientation of the region affect the limits of integration?

Tip: When dealing with transformations, always visualize the transformed region to ensure correct limits for integration.