Problem 12 - Basic Technical Mathematics with Calculus

Question

Calculate the indicated areas. All data are accurate to at least two significant digits. The widths (in \(\mathrm{m}\) ) of half the central arena in the Colosseum in Rome are shown in the following table, starting at one end and measuring from the middle to one side at \(4.0 \mathrm{m}\) intervals. Find the area of the arena by the trapezoidal rule. $$\begin{array}{lrrrrrr}\text { dist. from middle (m) } & 0.0 & 4.0 & 8.0 & 12.0 & 16.0 & 20.0 \\\\\text { width (m) } & 55.0 & 54.8 & 54.0 & 53.6 & 51.2 & 49.0 \\\\\text { dist. } & 24.0 & 28.0 & 32.0 & 36.0 & 40.0 & 44.0 \\\\\text { width } & 45.8 & 42.0 & 37.2 & 31.1 & 21.7 & 0.0 \end{array}$$

Step-by-Step Solution

Answer

The area is approximately 1871.6 square meters.

Step 1: Understanding the Trapezoidal Rule

The trapezoidal rule is a method to approximate the definite integral (or the area under a curve) by dividing the area into trapezoids rather than rectangles. The formula for the trapezoidal rule is given by:\[ A \approx \frac{h}{2} [f(x_0) + 2f(x_1) + 2f(x_2) + \cdots + 2f(x_{n-1}) + f(x_n)] \]where \( h \) is the distance between each point (interval width) and \( f(x_i) \) represents the function values (widths in this context). The ends are taken once, and all intermediate points are taken twice.

Step 2: Listing Known Data

From the table, we have the following distance-width pairs: \[(0.0, 55.0), (4.0, 54.8), (8.0, 54.0), (12.0, 53.6), (16.0, 51.2), (20.0, 49.0), (24.0, 45.8), (28.0, 42.0), (32.0, 37.2), (36.0, 31.1), (40.0, 21.7), (44.0, 0.0)\] The interval width \( h \) is consistently \(4.0 \) m between measured widths.

Step 3: Applying the Trapezoidal Rule

According to the trapezoidal rule formula: \[A \approx \frac{4.0}{2} \left[ 55.0 + 2(54.8) + 2(54.0) + 2(53.6) + 2(51.2) + 2(49.0) + 2(45.8) + 2(42.0) + 2(37.2) + 2(31.1) + 2(21.7) + 0.0 \right]\]\[ A \approx 2.0 \left[ 55.0 + 2(54.8 + 54.0 + 53.6 + 51.2 + 49.0 + 45.8 + 42.0 + 37.2 + 31.1 + 21.7) \right] \]

Step 4: Calculating the Sum Inside the Brackets

First, calculate the sum of all the widths, considering the multipliers: \[55.0 + 2(54.8 + 54.0 + 53.6 + 51.2 + 49.0 + 45.8 + 42.0 + 37.2 + 31.1 + 21.7)\]\[= 55.0 + 2(440.4) = 55.0 + 880.8 = 935.8\]

Step 5: Finding the Final Area

Finally, multiply this sum by 2 (from Step 3):\[A = 2.0 \times 935.8 = 1871.6 \text{ square meters} \]

Step 6: Conclusion

Hence, the estimated area of half the central arena in the Colosseum calculated using the trapezoidal rule is approximately 1871.6 square meters.

Key Concepts

Definite Integral

A definite integral represents the exact area under a curve within certain limits, offering a precise calculation of this area. When you see the mathematical notation \( \int_{a}^{b} f(x) \, dx \), this is describing a definite integral between the points \( a \) and \( b \) on the \( x \)-axis. This concept is crucial in many fields like physics and engineering, where it helps determine quantities like displacement, area, and total accumulated quantities over a continuous range.
Definite integrals provide a way to aggregate information from a function over an interval.

• Limits of Integration: \( a \) and \( b \) define the start and end points of the range.
• Function Under the Curve: \( f(x) \) is the function we are integrating.
• Result: The integration gives a numeric value representing the total area under \( f(x) \) from \( a \) to \( b \).

In scenarios where the integral cannot be computed analytically due to a complex function or lacking a closed form, numerical methods such as the Trapezoidal Rule become very handy. They enable a practical approach to estimating the area.

Numerical Integration

Numerical Integration is the practice of estimating the value of a definite integral when an exact solution is unrealistic or impossible to obtain. This often involves using methods such as the Trapezoidal Rule, Simpson's Rule, or constructing Riemann sums.
These techniques approximate the integral by summing up areas of geometrical shapes like trapezoids or rectangles under a curve.

• Discrete Sampling: Values of the function \( f(x) \) are used at specific points to calculate estimates.
• Approximation Techniques: These include dividing the function into equal segments and calculating each segment separately.
• Accuracy vs. Complexity: More points or complex shapes like parabolas can improve accuracy but at the cost of increased computational effort.

The Trapezoidal Rule, covered in the problem, efficiently uses trapezoids to form an approximation. It is particularly effective for functions that don't have dramatic changes in slope over the intervals. This method is very useful in real-world applications where sampling data at points is more feasible than obtaining the exact function value.

Approximation Methods

Approximation methods like the Trapezoidal Rule play a pivotal role in calculating integrals when direct analytical solutions are elusive. These methods simplify complex curves into easier-to-calculate shapes.
The rationale behind such methods is solid: if you can't find the exact area, estimate it closely.

• Trapezoidal Rule: This method divides the area under a curve into a series of trapezoids, summing their areas for an overall estimate. It's ideal when you have tabulated data, as in the Colosseum exercise.
• Function Approximation: By approximating \( f(x) \) as a series of straight-line segments or simple geometric shapes.
• Calculation Benefits: They provide flexibility and precision control, often adjustable by changing the number of divisions (trapezoids).

In the exercise, the consistent interval width simplifies calculations. Each added trapezoid increases precision, and though not perfect, the Trapezoidal Rule provides a practical and often sufficiently accurate solution for many real-world problems.