To approximate the area of the shaded region using the Trapezoidal Rule with $n = 4$, let's outline the process based on the given information.

Steps for Using the Trapezoidal Rule

The Trapezoidal Rule for estimating the area under a curve $y = f(x)$ over an interval $[a, b]$ 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:

• $n$ is the number of subintervals.
• $\Delta x = \frac{b - a}{n}$ is the width of each subinterval.
• $x_0, x_1, \ldots, x_n$ are the endpoints of the subintervals.

Given Information

• $n = 4$
• The interval $[a, b]$ appears to span approximately from $x = -20$ to $x = 20$.

Step 1: Determine $\Delta x$

1. Since $a = -20$ and $b = 20$, the interval width is: $\Delta x = \frac{20 - (-20)}{4} = \frac{40}{4} = 10$

2. This means we will evaluate the function at $x = -20, -10, 0, 10,$ and $20$.

Step 2: Approximate Function Values at the Points

Based on the graph:

• $f(-20) \approx 10$
• $f(-10) \approx 35$
• $f(0) \approx 40$
• $f(10) \approx 30$
• $f(20) \approx 15$

Step 3: Apply the Trapezoidal Rule Formula

Substitute these values into the Trapezoidal Rule formula:

$\int_{-20}^{20} f(x) \, dx \approx \frac{10}{2} \left[ f(-20) + 2f(-10) + 2f(0) + 2f(10) + f(20) \right]$

Calculating further:

1. $\frac{10}{2} = 5$
2. Substitute the function values: $= 5 \left[ 10 + 2(35) + 2(40) + 2(30) + 15 \right]$
3. Simplify inside the brackets: $= 5 \left[ 10 + 70 + 80 + 60 + 15 \right]$
4. Add the terms: $= 5 \times 235 = 1175$

Final Answer

The approximate area of the shaded region is 1175 square units.

Would you like a more detailed explanation on any step, or do you have further questions?

Here are some related questions:

1. How would the approximation change if $n$ were increased to 8?
2. What other methods besides the Trapezoidal Rule can be used to approximate areas?
3. How does the accuracy of the Trapezoidal Rule compare to Simpson's Rule?
4. What are the effects of increasing $n$ on the approximation error in numerical integration?
5. Can the Trapezoidal Rule be applied to functions that are not continuous?

Tip: Increasing the number of trapezoids (larger $n$) typically improves the approximation accuracy in the Trapezoidal Rule.