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The image shows the following double integration problem:

\[ z = 6 - 2y \]

\[ 0 \leq x \leq 4 \]
\[ 0 \leq y \leq 2 \]

The goal is to evaluate the double integral of the function \( z = 6 - 2y \) over the given rectangular region.

### Step-by-Step Solution

1. **Set Up the Double Integral:**

   The double integral is set up as:

   \[
   \iint_R z \, dA = \int_{x=0}^{4} \int_{y=0}^{2} (6 - 2y) \, dy \, dx
   \]

2. **Integrate with Respect to \( y \):**

   Integrate the inner integral first (with respect to \( y \)):

   \[
   \int_{y=0}^{2} (6 - 2y) \, dy
   \]

   Find the antiderivative of \( 6 - 2y \) with respect to \( y \):

   \[
   \text{Antiderivative} = 6y - y^2
   \]

   Now, evaluate this from \( y = 0 \) to \( y = 2 \):

   \[
   \left[ 6y - y^2 \right]_0^2 = (6(2) - 2^2) - (6(0) - 0^2) = (12 - 4) - 0 = 8
   \]

3. **Integrate with Respect to \( x \):**

   Now integrate the result with respect to \( x \):

   \[
   \int_{x=0}^{4} 8 \, dx
   \]

   The antiderivative of \( 8 \) with respect to \( x \) is \( 8x \). Evaluate this from \( x = 0 \) to \( x = 4 \):

   \[
   \left[ 8x \right]_0^4 = 8(4) - 8(0) = 32 - 0 = 32
   \]

4. **Final Answer:**

   The value of the double integral is \( 32 \).

Would you like further details or have any questions?  

Here are some related questions you might consider:
1. What happens if the region of integration is not rectangular?
2. How does changing the order of integration affect the result?
3. Can you evaluate a similar double integral if \( z = x + y \)?
4. What is the geometric interpretation of this double integral?
5. How would the result change if the limits for \( y \) were different?
6. Can this integral be evaluated using polar coordinates?
7. What is the physical significance of this double integral in real-world applications?
8. How does the value of the integral change if the function inside the integral is altered to \( z = 6 - 2y + x \)?

**Tip:** When evaluating double integrals, always verify the limits of integration to ensure they align with the region of interest.

Learn how to evaluate the double integral of the function z = 6 - 2y over a specified rectangular region. Follow a detailed step-by-step solution covering double integration and antiderivatives. Suitable for advanced high school students and beyond.

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