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Let's solve the given problems step-by-step.

### Problem (a): Estimating the Integral using the Midpoint Rule

Given:
- A contour map of a function \( f(x, y) \) on the square \( R = [0, 4] \times [0, 4] \).
- Use the Midpoint Rule with \( m = n = 2 \) to estimate the value of the double integral:

\[
\int \int_{R} f(x, y) \, dA
\]

#### Step-by-Step Solution:

1. **Divide the Region \( R \) into Equal Subrectangles:**

   Since \( m = n = 2 \), the region \( R = [0, 4] \times [0, 4] \) is divided into a 2x2 grid. Each subrectangle will have dimensions:
   \[
   \Delta x = \frac{4 - 0}{2} = 2, \quad \Delta y = \frac{4 - 0}{2} = 2.
   \]

2. **Determine the Midpoints of Each Subrectangle:**

   The midpoints of the 2x2 grid are:
   - Midpoint 1: \( (1, 1) \)
   - Midpoint 2: \( (3, 1) \)
   - Midpoint 3: \( (1, 3) \)
   - Midpoint 4: \( (3, 3) \)

3. **Estimate Function Values at Midpoints:**

   From the contour map, we estimate the values of \( f(x, y) \) at each midpoint:
   - \( f(1, 1) \approx 10 \)
   - \( f(3, 1) \approx 40 \)
   - \( f(1, 3) \approx 30 \)
   - \( f(3, 3) \approx 60 \)

4. **Apply the Midpoint Rule Formula:**

   The Midpoint Rule for double integrals is:
   \[
   \int \int_{R} f(x, y) \, dA \approx \Delta x \, \Delta y \sum f(x_i, y_j),
   \]
   where \( (x_i, y_j) \) are the midpoints.

   Substituting the values:
   \[
   \int \int_{R} f(x, y) \, dA \approx (2)(2) [f(1, 1) + f(3, 1) + f(1, 3) + f(3, 3)].
   \]

   Plugging in the function values:
   \[
   \int \int_{R} f(x, y) \, dA \approx 4 \times (10 + 40 + 30 + 60) = 4 \times 140 = 560.
   \]

   **Answer to (a):** The estimated value of the integral is **560**.

### Problem (b): Estimating the Average Value of \( f \)

To find the average value of \( f \) over the region \( R \), use the formula:

\[
\text{Average value of } f = \frac{1}{\text{Area of } R} \int \int_{R} f(x, y) \, dA.
\]

The area of \( R = [0, 4] \times [0, 4] \) is:
\[
\text{Area of } R = 4 \times 4 = 16.
\]

Substitute the estimated integral value from part (a):
\[
\text{Average value of } f = \frac{1}{16} \times 560 = 35.
\]

**Answer to (b):** The estimated average value of \( f \) is **35.0**.

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Would you like more details, or do you have any questions about these steps?

### Related Questions:
1. How would the estimate change if we used \( m = n = 4 \) for the Midpoint Rule?
2. What is the impact of contour spacing on the accuracy of the Midpoint Rule estimation?
3. How does the Midpoint Rule compare to other numerical integration methods like the Trapezoidal Rule?
4. How can the average value of \( f \) be interpreted in a real-world context?
5. How would using more subdivisions affect the estimated integral?

#### Tip:
When using the Midpoint Rule, remember that more subdivisions generally lead to a more accurate estimate of the integral.

A contour map is shown for a function f on the square R = [0, 4] x [0, 4]. Use the Midpoint Rule with m = n = 2 to estimate the value of the double integral ∫∫_R f(x, y) dA. Estimate the average value of f.

Learn how to use the Midpoint Rule to estimate the value of a double integral using a contour map for a function on the square R = [0, 4] x [0, 4]. Step-by-step explanation of finding the integral and average value of f.

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