https://file.aiquickdraw.com/mathbot/file/1720937535524u43qvhcy.png

Let's approach this problem step-by-step by correctly setting up the integration with the given transformations. 

Given:
\[ u = 4x + y \]
\[ v = -3x + 3y \]

Let's find the Jacobian determinant and the new bounds for \( u \) and \( v \).

The boundaries are given by:
1. \( 4x + y = 0 \)  --> \( u = 0 \)
2. \( 4x + y = 6 \)  --> \( u = 6 \)
3. \( -3x + 3y = 1 \)  --> \( v = 1 \)
4. \( -3x + 3y = 3 \)  --> \( v = 3 \)

We need to set up the integral in the new coordinates.

First, compute the Jacobian determinant:
\[
\begin{vmatrix}
\frac{\partial u}{\partial x} & \frac{\partial u}{\partial y} \\
\frac{\partial v}{\partial x} & \frac{\partial v}{\partial y}
\end{vmatrix}
=
\begin{vmatrix}
4 & 1 \\
-3 & 3
\end{vmatrix} = (4 \cdot 3) - (1 \cdot -3) = 12 + 3 = 15
\]

The integrand is \( \frac{u}{v} \), and we include the Jacobian determinant in the integration.

Let's set up the correct integral:

\[ \int_{v=1}^{v=3} \int_{u=0}^{u=6} \frac{u}{v} \cdot 15 \, du \, dv \]

Now let's calculate this integral step-by-step.The value of the integral is \( 296.6253 \) (rounded to four decimal places).

Do you want details or have any questions?

Here are 5 numbered questions you might want to ask:
1. Can you show the detailed steps of how to find the Jacobian determinant?
2. How did you determine the new bounds for \( u \) and \( v \)?
3. Can you explain why the integrand includes the Jacobian determinant?
4. How would the integral change if the bounds or the integrand were different?
5. Can you show the step-by-step process of performing the integration?

**Tip:** When solving problems involving change of variables, always carefully determine the new bounds and correctly calculate the Jacobian determinant to ensure accurate results.

Explore an advanced mathematical problem involving double integrals and the change of variables technique using the Jacobian determinant. Learn how to set up and solve integrals in new coordinate systems. Suitable for advanced undergraduate or graduate-level students.

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