Problem 22 - 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. \(^{2}\). Find the area of the circle using Simpson's rule and all values in the table. Explain why the value found is closer to 3.14 in. \(^{2}\) than the value found in Exercise 21.

Step-by-Step Solution

Answer

Using Simpson's rule, the area is approximately 2.598 in.², a closer estimate to 3.14 in.² due to better curve approximation.

Step 1: Understanding Simpson's Rule

Simpson's rule is a method for numerical integration. It approximates the integral of a function by dividing the area under the curve into an even number of subintervals and fitting parabolas to each pair of subintervals. The formula for Simpson's rule is \( \frac{h}{3} \left [ f(x_0) + 4 \sum_{i=1,3,5,...}^{n-1} f(x_i) + 2 \sum_{i=2,4,6,...}^{n-2} f(x_i) + f(x_n) \right ] \), where \( h \) is the distance between subintervals, \( n \) is even, and \( f(x_i) \) are the function values.

Step 2: Identify Variables

We know that the diameter of the circle is 2 inches, so the distance \( h \), which is the uniform spacing between the chords, is 0.250 inches as given from the distance table. The function values \( f(x_i) \) are the lengths of the chords provided in the table.

Step 3: Apply Simpson's Rule

We have 9 points (from \( x_0 \) to \( x_8 \)) in the table, which creates 8 intervals (an even number). Applying Simpson’s rule, calculate:\[\frac{0.250}{3} \left [ 0 + 4(1.323 + 1.936 + 1.936 + 1.323) + 2(1.732 + 1.732) + 0 \right ]\]This evaluates to:\[\frac{0.250}{3} \left [ 15.592 \right ] = 1.299\]

Step 4: Integrate and Adjust

The area found using Simpson's rule is for half of the circle since the chord lengths given in the table cover from one diameter end to the center and back. Therefore, multiply by 2:\[2 \times 1.299 = 2.598\]

Step 5: Comparison and Conclusion

The calculated area using Simpson's rule is closer to the theoretical area of the circle (3.14 \( \text{in}^2 \)) because Simpson's Rule accounts for the curvature of the circle through quadratic approximations. The precise match is not exact because of the sampling (discrete nature) and approximations inherent in Simpson's rule, but it is a closer estimate compared to a simpler method such as using trapezoids.

Key Concepts

Numerical Integration

Numerical integration is an essential technique in calculus that provides approximate solutions to definite integrals. Often, when dealing with integrals that are difficult or impossible to solve algebraically, numerical methods like Simpson's Rule become extremely valuable.

Numerical integration works by breaking down a function's curve into smaller, more manageable sections. This segmentation allows for approximating the area under the curve. Think of it as piecing together slices to form an approximate whole area. The two most commonly used methods in numerical integration are the Trapezoidal Rule and Simpson's Rule.

Simpson's Rule takes this process a step further by using parabolas instead of straight lines (as in the Trapezoidal Rule) to create a more precise estimate. This method fits quadratic functions across pairs of points, effectively smoothing out the curve and providing a more accurate approximation for complex shapes like curves.

Given its sophistication, Simpson's Rule is particularly suited for functions with regular, smooth curves over the integration interval. This adaptability is why, as seen in our original exercise, Simpson's Rule delivers a closer estimate to the circle's true area.

Circle Area Calculation

Circle area calculation is a common problem often solved in many introductory mathematics courses. The formula for the area of a circle, which is \(A = \pi r^2\), gives the exact area for circles of any size, where \(r\) denotes the radius. However, calculating the area of a circle using data points and numerical techniques offers practical space to apply numerical integration.

In situations where only discrete data points are available, like a list of chord lengths at specific intervals across a circle's diameter, applying a method like Simpson's Rule provides an alternate approach to calculating area.

• First, identify the uniform interval between data points. In our exercise, this is 0.250 inches.
• Use the given chord lengths to apply Simpson's approximation.

By calculating the area over half of the circle from these sampled chord lengths and then doubling the result, we can approximate the full circle's area. Though computationally more elaborate than the straightforward \(A = \pi r^2\), this method aptly uses sampled data to give a good prediction of area.

Calculus Methods

Calculus methods, including differentiation and integration, are foundational for understanding and working with the changing quantities within mathematics. Integration, in particular, finds the total amount of something given its rate of change, which can be visualized as finding the space under a curve.

While basic calculus often handles problems using symbols and algebraic manipulations, the real world regularly demands numeric solutions where calculus methods must adapt to the available data. This adaptation is where numerical approaches, like Simpson's Rule, come into play. They allow for effective calculation of areas, volumes, and other integrals when exact solutions are hard to come by.

In applying these methods, one must:

• Choose the suitable numerical method based on the data and function shape.
• Implement the process using the given intervals and function values.

By turning intricate continuous curves into a set of discrete data points and approximations, numerical calculus methods expand the practicality of calculus into more applicable, tangible problem-solving tools such as the circle area calculation seen in our exercise, highlighting their importance in numerical mathematics.