Problem 17 - Basic Technical Mathematics with Calculus

Question

calculate the area of the circle by the indicated method. The lengths of parallel chords of a circle that are 0.250 in. apart are given in the following table. The diameter of the circle is 2.000 in. The distance shown is the distance from one end of a diameter. $$\begin{array}{l|l|l|l|l|l|l|l} \text {Distance (in.) } & 0.000 & 0.250 & 0.500 & 0.750 & 1.000 & 1.250 & 1.500 & 1.750 & 2.000 \\ \hline \text {Length (in.) } & 0.000 & 1.323 & 1.732 & 1.936 & 2.000 & 1.936 & 1.732 & 1.323 & 0.000 \end{array}$$ Using the formula \(A=\pi r^{2},\) the area of the circle is 3.14 in. Find the area of the circle using the trapezoidal rule and only the values of distance of 0.000 in., 0.500 in. 1.000 in., 1.500 in., and 2.000 in. with the corresponding values of the chord lengths. Explain why the value found is less than 3.14 in. \(^{2}\)

Step-by-Step Solution

Answer

Using the trapezoidal rule, the area is approximately 2.732 in², which is less than 3.14 in² due to underestimation of curvature.

Step 1: Understanding the Problem

We need to find the area of a circle using the trapezoidal rule based on given chord lengths at certain distances from a diameter. The diameter of the circle is 2 inches, implying a radius of 1 inch. These chord lengths approximate the width of segments of the circle at specific intervals.

Step 2: Select Data Points

From the table, select the given distances: 0.000, 0.500, 1.000, 1.500, and 2.000 inches. Corresponding lengths of the chords are: 0.000, 1.732, 2.000, 1.732, and 0.000 inches, respectively.

Step 3: Apply the Trapezoidal Rule

The trapezoidal rule is used to approximate the integral of a function. In this context, it is: \[ A \approx \frac{d}{2} (L_0 + 2L_1 + 2L_2 + 2L_3 + L_4) \]where \( d \) is the distance between the given points, 0.5 inches, and \( L_i \) are the chord lengths. Substitute known values: \[ A \approx \frac{0.5}{2} (0.000 + 2 \times 1.732 + 2 \times 2.000 + 2 \times 1.732 + 0.000) \].

Step 4: Perform the Calculation

Substitute the values into the formula: \[ A \approx 0.25 (0.000 + 3.464 + 4.000 + 3.464 + 0.000) \]\[ A \approx 0.25 \times 10.928 \]\[ A \approx 2.732 \text{ in}^2 \].

Step 5: Discuss the Result

The calculated area using the trapezoidal rule, 2.732 square inches, is less than the actual area 3.14 square inches because the trapezoidal rule underestimates the true area when the function (circle in this case) is curved and the approximation intervals are linear. The trapezoids do not completely cover the curvature of the circle.

Key Concepts

Trapezoidal Rule

The trapezoidal rule is a numerical method used to estimate the integral, or the area under a curve. It simplifies the process by dividing the area into trapezoids instead of rectangles. This method is particularly helpful when dealing with non-linear shapes, such as curves.
The formula for the trapezoidal rule is: \[ A \approx \frac{d}{2} \left(L_0 + 2L_1 + 2L_2 + \ldots + 2L_{n-1}+ L_n\right) \]

• \(d\) is the distance between data points. In this example, it is 0.5 inches.
• \(L_i\) represents the lengths of the chords at these distances.

By substituting the values into this formula, we can find an estimated area under the curve described by the chord data. This method is practical but has limitations, especially when approximating highly curved shapes like circles.

Circle Area Calculation

To calculate the circle's area using conventional methods, we apply the formula \(A = \pi r^2\), where \(r\) is the radius of the circle. For our problem, the circle has a diameter of 2 inches, meaning its radius is 1 inch.
Thus, the area could be computed as:\[ A = \pi \times (1)^2 = \pi \approx 3.14\text{ in}^2 \].

• We use \(\pi\) in this formula as it accounts for the curved nature of the circle.
• Each unit, both radius and area, is represented in inches, which provides a consistent measurement basis.

Compared to our trapezoidal estimate, this calculation is more accurate, highlighting the discrepancy due to the curvature of the circle.

Chord Lengths

Chord lengths are critical to our exercise, where the circle's curve is approximated by straight line segments. A chord is a line segment whose endpoints both lie on the circle.
We use these lengths at various points to estimate the circle's area indirectly.
Here are some important points about chord lengths in this context:

• Chords are parallel and equidistant, positioned 0.250 inches apart.
• They vary in length as they move across the circle, achieving the longest length at the circle's center.
• The symmetry of chord lengths relative to the circle’s center helps improve the accuracy of the trapezoidal approximation.

Understanding how these chord lengths change and where they are placed is essential to setting up the trapezoidal rule correctly.

Integral Approximation

Integral approximation is a vital concept in calculus that involves estimating the value of an integral or the total area under a curve. When we apply this concept using numerical methods like the trapezoidal rule, we work to find a sum that resembles the actual integral.
Here are a few key points:

• This approximation is essential for curves, like our circle, where traditional integration may be complex or infeasible directly.
• Methods like the trapezoidal rule provide a simpler way to estimate areas, though they can lead to underestimation if intervals are too wide or the curve is too complex.
• With a smooth curve, more data points distributed evenly will often yield a more accurate integral approximation.

Overall, integral approximation simplifies working with curves, assisting in translating complex figures into comprehensible calculations.