To approximate the area of the shaded region using the Trapezoidal Rule with $n = 4$, we need to:

1. Divide the interval into $n = 4$ subintervals of equal width.
2. Apply the Trapezoidal Rule formula: $A \approx \frac{\Delta x}{2} \left[ f(x_0) + 2f(x_1) + 2f(x_2) + \dots + 2f(x_{n-1}) + f(x_n) \right],$ where $\Delta x = \frac{b-a}{n}$, and $f(x_i)$ are the values of the function at each interval point.

Steps:

1. Determine the interval and width ($\Delta x$):

   • From the graph, the interval is approximately $[-20, 20]$.
   • Subdivide this interval into $n = 4$: $\Delta x = \frac{20 - (-20)}{4} = \frac{40}{4} = 10.$

   The points are $x_0 = -20$, $x_1 = -10$, $x_2 = 0$, $x_3 = 10$, and $x_4 = 20$.

2. Read $f(x)$ values from the graph: Estimate the height of the function at each point:

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

3. Apply the Trapezoidal Rule: Substitute the values into the formula: $A \approx \frac{\Delta x}{2} \left[ f(x_0) + 2f(x_1) + 2f(x_2) + 2f(x_3) + f(x_4) \right].$ With $\Delta x = 10$: $A \approx \frac{10}{2} \left[ 10 + 2(30) + 2(40) + 2(30) + 20 \right].$

   Simplify: $A \approx 5 \left[ 10 + 60 + 80 + 60 + 20 \right].$ $A \approx 5 \times 230 = 1150.$

Final Answer:

The approximate area of the shaded region is $\mathbf{1150}$ square units.

Would you like more details on the method or assumptions? Here are five related questions to explore further:

1. How does increasing $n$ affect the accuracy of the Trapezoidal Rule?
2. What are the limitations of using the Trapezoidal Rule for curved regions?
3. How does the Trapezoidal Rule compare to Simpson’s Rule in terms of accuracy?
4. Can we verify this approximation with exact integration methods if the function is known?
5. How would errors in reading $f(x)$ values from the graph affect the result?

Tip: Always try to increase the number of subintervals for a more accurate approximation, especially for functions with rapid changes!