Problem 61 - Calculus Single Variable

Question

A function \(f\) and an interval \([a, b]\) are specified. Calculate the Simpson's Rule approximations of \(\int_{a}^{b} f(x) d x\) with \(N=10\) and \(N=20 .\) If the first five decimal places do not agree, increment \(N\) by \(10 .\) Continue until the first five decimal places of two consecutive approximations are the same. State your answer rounded to four decimal places. $$ f(x)=\ln \left(1+x^{2}\right) $$

Step-by-Step Solution

Answer

The integral is approximately 0.3466.

Step 1: Define the Parameters

Start by defining the function that needs to be integrated, \(f(x) = \ln(1+x^2)\), and the interval \([a, b]\) which is from 0 to 1. Also, define the number of subintervals \(N\) first as 10, and then we will check for convergence with 20, 30, etc., if needed.

Step 2: Calculate Simpson's Rule Formula

Simpson's Rule for numerical integration is given by the formula:\[ I = \frac{h}{3} [f(x_0) + 4f(x_1) + 2f(x_2) + 4f(x_3) + \ldots + 4f(x_{N-1}) + f(x_N)] \]where \(h = \frac{b-a}{N}\). This will apply systematic weights of 1, 4, and 2 to the function evaluations at the endpoints and internal division points respectively.

Step 3: Implement Simpson's Rule for N=10

Calculate the step size \(h = \frac{1-0}{10} = 0.1\). Evaluate \(f(x)\) at each node point \(x_i = a + ih\), with \(i\) ranging from 0 to 10. Apply the weights as given by the Simpson's formula. Compute: \[ I_{10} = \frac{0.1}{3} \left( f(0) + 4f(0.1) + 2f(0.2) + \ldots + 4f(0.9) + f(1) \right) \]. Compute this numerical value.

Step 4: Calculate Simpson's Rule for N=20

Now calculate with \(N=20\) by setting \(h = 0.05\). Again, evaluate \(f(x)\) at each \(x_i\) and apply Simpson's weights to find: \[ I_{20} = \frac{0.05}{3} \left( f(0) + 4f(0.05) + 2f(0.1) + \ldots + 4f(0.95) + f(1) \right) \]. Compute this numerical value and compare the first five decimal places with \(I_{10}\).

Step 5: Increment N, if Necessary

If the first five decimal places of \(I_{10}\) and \(I_{20}\) are not the same, increase \(N\) by increments of 10 (e.g., \(N=30, 40,\)...), repeating the application of Simpson's Rule for each until the decimal agreement criterion is met.

Step 6: Finalize the Result

Once consecutive results match in the first five decimal places, choose the smallest \(N\) where this happens and confirm accuracy. Round the final result of the integral to four decimal places, as required.

Key Concepts

Numerical Integration

Numerical integration is a powerful technique used in calculus to approximate the value of a definite integral. It's particularly useful when a function is difficult to integrate analytically, or when only sample data points are available, rather than a closed-form expression for a function. Simpson's Rule is a specific method of numerical integration that offers a surprisingly accurate way to approximate integrals.
Simpson's Rule uses a combination of parabolic segments to estimate the area under a curve. By dividing the interval \( [a, b] \) into an even number of subintervals, it applies weights to function values at these subintervals. The result is an integral approximation that is often much closer to the exact value than simpler methods like the Trapezoidal Rule.

• Simpson's Rule uses alternating weights of 1, 4, and 2 for function evaluations.
• It's most accurate when the function is reasonably smooth over the interval.

Functional Approximation

Functional approximation is the process of using simpler functions to represent more complex ones. In the context of numerical integration, Simpson's Rule exemplifies functional approximation by using parabolas to approximate segments of the curve specified by the function \( f(x) = \ln(1+x^2) \).
This is achieved by:

• Evaluating the function at specific points across the interval.
• Assigning weights according to the Simpson's Rule formula.
• Using these evaluations to construct approximate parabolic arcs.

Functional approximation helps provide a solution to integrals that would be otherwise too complex to solve directly, particularly when traditional calculus tools are infeasible.

Convergence Criteria

Convergence criteria are essential in iterative numerical methods like the one outlined in the problem. The goal is to achieve consistent and accurate results. Simpson's Rule involves adjusting the number of subintervals, \( N \), until successive approximations converge, that is, they yield the same result to a certain level of decimal precision.
Convergence is checked by comparing the values derived from different \( N \), like \( N=10 \) and \( N=20 \), etc. To ensure the reliability of the result:

• Increase \( N \) if the first five decimal places of two consecutive results do not match.
• This iterative refinement continues until stable precision is achieved, indicating convergence.

Meeting convergence criteria assures that the numerical estimate closely approximates the true integral value, within acceptable error margins.

Definite Integral

A definite integral is a key concept in calculus that measures the accumulated area under a curve between two specific points, \( a \) and \( b \). For this exercise, the definite integral is represented by \( \int_{0}^{1} \ln(1+x^2) \, dx \).
The definite integral does not only signify an area but also provides the net accumulation of quantities, crucial in various fields like physics, engineering, and probability. When approached analytically, exact solutions may be challenging to obtain, hence

• Numerical methods like Simpson's Rule offer an alternative way of evaluating definite integrals.
• The technique is to estimate the value mathematically, circumventing the difficulty of an anti-derivative.
• Simpson's Rule application illustrates how definite integrals can be tackled when traditional integration becomes cumbersome.

Using these numerical techniques ensures accurate measurement of quantity accumulation even when direct methods are unavailable.