Problem 11 - Basic Technical Mathematics with Calculus

Question

Calculate the indicated areas. All data are accurate to at least two significant digits. Using aerial photography, the widths of an area burned by a forest fire were measured at \(0.5-\mathrm{mi}\) intervals, as shown in the following table: $$\begin{array}{l|c|c|c|c|c|c|c|c|c}\text {Distance (mi)} & 0.0 & 0.5 & 1.0 & 1.5 & 2.0 & 2.5 & 3.0 & 3.5 & 4.0 \\\\\hline \text {Width (mi)} & 0.6 & 2.2 & 4.7 & 3.1 & 3.6 & 1.6 & 2.2 & 1.5 & 0.8\end{array}$$ Determine the area burned by the fire by using the trapezoidal rule.

Step-by-Step Solution

Answer

The area burned by the fire is approximately 9.8 square miles.

Step 1: Understand the Trapezoidal Rule

The Trapezoidal Rule is a numerical method to approximate the definite integral of a function. When given a set of points, it works by approximating the area under the curve as a series of trapezoids and summing their areas. The formula for the Trapezoidal Rule is: \[ A \approx \frac{h}{2} \left( f(x_0) + 2f(x_1) + 2f(x_2) + \cdots + 2f(x_{n-1}) + f(x_n) \right) \] where \( h \) is the interval width and \( f(x_i) \) are the function values at each interval. In this case, \( h = 0.5 \; \text{mi} \).

Step 2: Identify Values

Identify the distances \( x_i \) and corresponding widths \( f(x_i) \) from the table: \( x_0 = 0.0 \), \( x_1 = 0.5 \), \( x_2 = 1.0 \), ..., \( x_8 = 4.0 \) and \( f(x_0) = 0.6 \), \( f(x_1) = 2.2 \), ..., \( f(x_8) = 0.8 \). The width at each distance is equivalent to evaluating the function \( f \) at that point.

Step 3: Apply Trapezoidal Rule Formula

Substitute the identified values into the trapezoidal rule formula: \[ A \approx \frac{0.5}{2} \left( 0.6 + 2(2.2) + 2(4.7) + 2(3.1) + 2(3.6) + 2(1.6) + 2(2.2) + 2(1.5) + 0.8 \right) \]

Step 4: Calculate Each Term

Calculate each term within the parenthesis: \[ 0.6 + 2(2.2) + 2(4.7) + 2(3.1) + 2(3.6) + 2(1.6) + 2(2.2) + 2(1.5) + 0.8 \] gives \[ 0.6 + 4.4 + 9.4 + 6.2 + 7.2 + 3.2 + 4.4 + 3.0 + 0.8 = 39.2 \]

Step 5: Final Calculation for Area

Now, multiply by \( \frac{0.5}{2} = 0.25 \): \[ A \approx 0.25 \times 39.2 = 9.8 \]

Step 6: Conclusion

The total area burned by the fire, as approximated using the trapezoidal rule, is \( 9.8 \; \text{mi}^2 \).

Key Concepts

Trapezoidal Rule

The Trapezoidal Rule is an essential numerical method often used in engineering, physics, and mathematics to estimate definite integrals. Rather than integrating a function directly, which can be complex, this method approximates the area under a curve by dividing it into several trapezoids.
This approach is particularly useful when dealing with data that consists of discrete points, like aerial photographs of a forest fire. Each trapezoid is formed between two successive data points. The width of these trapezoids is determined by the interval between consecutive points, denoted as \( h \). For each interval, the area is calculated and then summed to find the total area under the curve.

In our example, the forest fire data is divided into intervals of \( 0.5 \; \text{mi} \), and the corresponding widths of the fire's reach are treated as function values. The trapezoidal rule formula applied here is:

• \( A \approx \frac{h}{2} (f(x_0) + 2f(x_1) + 2f(x_2) + \ldots + 2f(x_{n-1}) + f(x_n)) \)

This equation efficiently sums up the areas of the trapezoids to arrive at an approximate integral, offering a practical alternative when the function is complex or known only at certain points.

Definite Integral Approximation

When calculating the area under a curve using the Definite Integral, we are essentially summing up an infinite number of infinitely small rectangles. However, since we often work with data points and do not always have the luxury of an explicit function, methods like the Trapezoidal Rule are invaluable.
The approximation of a definite integral is achieved by summing the area of trapezoids across the interval of interest. This is particularly beneficial when dealing with real-world situations, like the forest fire scenario, where precise measurements may only be available at discrete intervals. Here, we take the sum of all the small trapezoidal slices to estimate the total area.

Firstly, by identifying function values \( f(x_i) \) at different points along the interval, we can utilize the widths of these intervals \( h \). As \( h \) approaches zero, the approximation becomes more precise, illustrating the relationship between numerical methods and analytical calculus. Using these concepts facilitates solving integrals in cases where expressing a function analytically is not possible.

Area Calculation

Calculating an area using the trapezoidal rule follows systematic steps, combining both numerical approximation and practical application. The purpose is to find the total area covered by a specific event, object, or phenomenon—in this case, the expanse affected by a forest fire.

Each trapezoid constructed has a height equal to the interval between measurements, and both bases aligned with the widths of the area at consecutive points. The formula for the area \( A \) of one trapezoid is:

• \( A = \frac{b_1 + b_2}{2} \times h \)

where \( b_1 \) and \( b_2 \) are the widths at each interval's beginning and end.
By applying this formula, multiplying by the sum of all trapezoidal areas, one obtains the total approximate area. Through a detailed breakdown, like substituting precise numeric values into the trapezoidal rule formula, and calculating step-by-step, we reach the conclusion: the burnt forest covers approximately \( 9.8 \; \text{mi}^2 \).

This methodical strategy not only enhances accuracy in practical situations but also serves as a pedagogical tool, illustrating how complex integrals can be solved with simpler, real-world data.