To approximate the area of the shaded region using the Trapezoidal Rule with $n = 4$, we'll follow these steps:

Step 1: Identify the interval and the number of trapezoids.

The graph shows a region between $x = -20$ and $x = 20$, so the interval is $[-20, 20]$.

Given $n = 4$, we will divide this interval into 4 subintervals, which means we need to calculate the step size $h$.

The formula for the step size is:

$h = \frac{b - a}{n}$

Where $a = -20$, $b = 20$, and $n = 4$. So,

$h = \frac{20 - (-20)}{4} = \frac{40}{4} = 10$

Thus, each subinterval will have a width of 10.

Step 2: Define the points.

The points where we will evaluate the function are: $x_0 = -20, \quad x_1 = -10, \quad x_2 = 0, \quad x_3 = 10, \quad x_4 = 20$

Step 3: Approximate the function values at these points.

From the graph, we estimate the function values at each $x_i$:

• $f(x_0) = f(-20) \approx 0$
• $f(x_1) = f(-10) \approx 30$
• $f(x_2) = f(0) \approx 40$
• $f(x_3) = f(10) \approx 30$
• $f(x_4) = f(20) \approx 0$

Step 4: Apply the Trapezoidal Rule formula.

The Trapezoidal Rule formula is:

$A \approx \frac{h}{2} \left[ f(x_0) + 2f(x_1) + 2f(x_2) + 2f(x_3) + f(x_4) \right]$

Substitute the values:

$A \approx \frac{10}{2} \left[ 0 + 2(30) + 2(40) + 2(30) + 0 \right]$

$A \approx 5 \left[ 0 + 60 + 80 + 60 + 0 \right]$

$A \approx 5 \times 200 = 1000$

Conclusion:

The approximate area of the shaded region using the Trapezoidal Rule with $n = 4$ is 1000 square units.

Would you like more details on any of the steps? Here are a few questions related to the topic:

1. How does increasing $n$ affect the accuracy of the Trapezoidal Rule?
2. What is the difference between the Trapezoidal Rule and Simpson's Rule?
3. Can we use the Midpoint Rule for the same problem? How would it differ?
4. What are the potential errors associated with the Trapezoidal Rule?
5. How can we derive the formula for the Trapezoidal Rule?

Tip: Using more subintervals (larger $n$) generally improves the approximation accuracy in numerical integration methods.