To solve the problem of approximating the area of the shaded region using the trapezoidal rule with $n = 4$, I will proceed as follows:

Trapezoidal Rule Formula

The trapezoidal rule for approximating an integral is given by: $\int_a^b f(x) \, dx \approx \frac{\Delta x}{2} \left[ f(x_0) + 2f(x_1) + 2f(x_2) + \cdots + 2f(x_{n-1}) + f(x_n) \right],$ where:

• $a$ and $b$ are the bounds of integration,
• $n$ is the number of subintervals,
• $\Delta x = \frac{b-a}{n}$,
• $x_i$ are the endpoints of each subinterval.

Steps to Solve:

1. Identify the bounds $a$ and $b$: From the graph, the bounds of the shaded region are $a = -20$ and $b = 20$.

2. Determine the subinterval width: With $n = 4$, $\Delta x = \frac{b-a}{n} = \frac{20 - (-20)}{4} = \frac{40}{4} = 10.$

3. Find the function values at the endpoints: Divide the interval into 4 equal parts: $x_0 = -20$, $x_1 = -10$, $x_2 = 0$, $x_3 = 10$, $x_4 = 20$. From the graph, approximate the values of $f(x)$ at these points: $f(-20) \approx 10, \, f(-10) \approx 30, \, f(0) \approx 40, \, f(10) \approx 30, \, f(20) \approx 20.$

4. Apply the trapezoidal rule: Substitute these values into the formula: $\int_{-20}^{20} f(x) \, dx \approx \frac{10}{2} \left[ f(-20) + 2f(-10) + 2f(0) + 2f(10) + f(20) \right].$ Simplify: $\int_{-20}^{20} f(x) \, dx \approx 5 \left[ 10 + 2(30) + 2(40) + 2(30) + 20 \right].$ $\int_{-20}^{20} f(x) \, dx \approx 5 \left[ 10 + 60 + 80 + 60 + 20 \right].$ $\int_{-20}^{20} f(x) \, dx \approx 5 \times 230 = 1150.$

Final Answer:

The approximate area of the shaded region is 1150 square units.

Let me know if you would like further clarification or additional details!

Here are five related questions to expand your understanding:

1. What is the error bound for the trapezoidal rule, and how can it be minimized?
2. How does increasing $n$ affect the accuracy of the trapezoidal rule?
3. What are the key differences between the trapezoidal rule and Simpson’s rule?
4. How is the integral approximation affected by uneven intervals of $x$?
5. Could we use numerical integration methods other than the trapezoidal rule for this problem?

Tip: Always try to double-check function values from the graph when using numerical methods, as slight inaccuracies can affect the final result.