To approximate the area of the shaded region using the Trapezoidal Rule with $n = 3$, we need to follow these steps:

Step 1: Define the Interval and Width of Each Subinterval

The graph shows a shaded area with an approximate range along the x-axis from about $x = -20$ to $x = 15$. Since $n = 3$, we will divide this interval into 3 equal subintervals.

1. Determine the total interval length: $15 - (-20) = 35$.
2. Calculate the width of each trapezoid, $\Delta x$: $\Delta x = \frac{35}{3} \approx 11.67$

Step 2: Identify Function Values at Each Interval

Using the approximate x-values at each subinterval, we can identify points $x_0, x_1, x_2,$ and $x_3$:

• $x_0 = -20$
• $x_1 \approx -8.33$
• $x_2 \approx 3.33$
• $x_3 = 15$

From the graph, estimate the function values $f(x)$ at these points. Here are approximate values based on visual inspection:

• $f(x_0) \approx 30$
• $f(x_1) \approx 18$
• $f(x_2) \approx 10$
• $f(x_3) \approx 25$

Step 3: Apply the Trapezoidal Rule

The Trapezoidal Rule formula for approximating the area $A$ is: $A \approx \frac{\Delta x}{2} \left[ f(x_0) + 2f(x_1) + 2f(x_2) + f(x_3) \right]$

Substitute the values: $A \approx \frac{11.67}{2} \left[ 30 + 2(18) + 2(10) + 25 \right]$ $= \frac{11.67}{2} \left[ 30 + 36 + 20 + 25 \right]$ $= \frac{11.67}{2} \times 111$ $= 11.67 \times 55.5 \approx 648.785$

Conclusion

The approximate area of the shaded region using the Trapezoidal Rule with $n = 3$ is $\approx 648.79$ square units.

Would you like further details on any part of this solution?

Here are five related questions to expand your understanding:

1. How does increasing the value of $n$ affect the accuracy of the Trapezoidal Rule?
2. How would the calculation change if we used Simpson's Rule instead of the Trapezoidal Rule?
3. What is the significance of the choice of points $x_0, x_1, x_2,$ and $x_3$ in the approximation?
4. How could you estimate the error of this approximation?
5. What are other numerical integration techniques that could be used for this problem?

Tip: Always double-check the function values from the graph to ensure accuracy in numerical approximations.