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:

\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.