https://file.aiquickdraw.com/mathbot/file/1731681673073p0ftbj8p.jpg

To approximate the area of the shaded region using the Trapezoidal Rule with \( n = 4 \), we follow these steps:

1. **Identify the Interval**: The graph spans approximately from \( x = -20 \) to \( x = 20 \), so the interval length is \( 40 \).

2. **Calculate \( \Delta x \)**: With \( n = 4 \) trapezoids, \( \Delta x = \frac{40}{4} = 10 \).

3. **Determine Function Values at Intervals**: We need to find the function values at \( x = -20 \), \( x = -10 \), \( x = 0 \), \( x = 10 \), and \( x = 20 \).
   - From the graph, approximate these values as:
     - \( f(-20) \approx 10 \)
     - \( f(-10) \approx 30 \)
     - \( f(0) \approx 40 \)
     - \( f(10) \approx 30 \)
     - \( f(20) \approx 10 \)

4. **Apply the Trapezoidal Rule Formula**:
   \[
   \text{Area} \approx \frac{\Delta x}{2} \left( f(x_0) + 2f(x_1) + 2f(x_2) + 2f(x_3) + f(x_4) \right)
   \]
   Substituting the values:
   \[
   \text{Area} \approx \frac{10}{2} \left( 10 + 2(30) + 2(40) + 2(30) + 10 \right)
   \]
   Simplify inside the parentheses:
   \[
   = 5 \left( 10 + 60 + 80 + 60 + 10 \right)
   \]
   \[
   = 5 \times 220 = 1100
   \]

So, the approximate area of the shaded region is \( 1100 \) square units.

Would you like more details or have any questions about the process?

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Here are some related questions to expand on this topic:

1. How would the approximation change if \( n = 8 \) trapezoids were used?
2. What are the potential errors involved in using the Trapezoidal Rule for approximation?
3. How does the Trapezoidal Rule compare to Simpson’s Rule for accuracy?
4. What is the exact area if we integrate the function analytically?
5. How would the approximation change if we used a smaller interval size (e.g., \( \Delta x = 5 \))?

**Tip**: The Trapezoidal Rule works better when the function is nearly linear over each interval. For highly curved functions, consider using more trapezoids or Simpson's Rule for greater accuracy.

Approximate the area of the shaded region by using the Trapezoidal Rule with n = 4.

Learn how to approximate the area of a shaded region using the Trapezoidal Rule with n = 4. This step-by-step guide covers calculating Δx, finding function values at intervals, and applying the Trapezoidal Rule formula, yielding an approximate area of 1100 square units. Suitable for high school calculus students.

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