Problem 66 - Calculus Single Variable

Question

An integral \(\int_{a}^{b} f(x) d x\) and a positive integer \(N\) are given. Compute the exact value of the integral, the Simpson's Rule approximation of order \(N,\) and the absolute error \(\varepsilon\). Then find a value \(c\) in the interval \((a, b)\) such that \(\varepsilon=(b-a)^{5}\left|f^{(4)}(c)\right| /\left(180 \cdot N^{4}\right) .\) (This form of the error, which resembles the Mean Value Theorem, implies inequality \((5.8 .4) .)\) $$ \int_{1}^{e} 1 / x d x \quad N=4 $$

Step-by-Step Solution

Answer

Exact integral value is 1; use Simpson’s Rule and derivative to find ε and c.

Step 1: Compute the Exact Value of the Integral

To compute the exact value of the integral \(\int_{1}^{e} \frac{1}{x} \, dx\), we recognize that the integral of \(\frac{1}{x}\) is \(\ln|x|\). Thus, we have:\[\int_{1}^{e} \frac{1}{x} \, dx = \left[ \ln|x| \right]_1^e = \ln(e) - \ln(1) = 1 - 0 = 1.\]

Step 2: Apply Simpson's Rule with N=4

Simpson's Rule states that the approximation of the integral over the interval \([a, b]\) can be computed as follows:\[S_N = \frac{b-a}{3N} \left( f(x_0) + 4 \sum_{i \text{ odd}} f(x_i) + 2 \sum_{i \text{ even}} f(x_i) + f(x_N) \right),\]where \(x_i = a + i\cdot h\) and \(h = \frac{b-a}{N}\). For \([1, e]\) with \(N=4\), \(h = \frac{e - 1}{4}\). Calculating each \(x_i\), we have \(x_0=1\), \(x_1 = 1 + h\), \(x_2\), \(x_3\), and \(x_4 = e\). Then compute and sum \(f(x_0), f(x_1), f(x_2), f(x_3), f(x_4) = \frac{1}{x_i}: \)\[f(1) = 1, \: f(e) = \frac{1}{e},\]summations for odd and even indices will follow from \(f(x_i).\) Finally, substitute into the Simpson's formula to get the approximation.

Step 3: Calculate the Absolute Error \(\varepsilon\)

Compute the absolute error \(\varepsilon = |\text{Exact Value} - \text{Simpson's Approximation}|\). Given exact calculation produces 1, and an approximation from Step 2, \(\varepsilon = |1 - \text{Simpson's Approximation}|\).

Step 4: Find c using Error Term Analysis

Using the error form \[\varepsilon = \frac{(b-a)^5}{180 \cdot N^4} \left|f^{(4)}(c)\right|,\] identify \(f^{(4)}(c)\). Since \(f = \frac{1}{x}\), derivatives can be computed:- \(f(x) = x^{-1}\)- \(f'(x) = -x^{-2}\)- \(f''(x) = 2x^{-3}\)- \(f'''(x) = -6x^{-4}\)- \(f^{(4)}(x) = 24x^{-5}\).Substitute numerical values to find \(c\) such that \(f^{(4)}(c)\) fits the error balance equation with computed \(\varepsilon\).

Key Concepts

Integral Calculus

Integral calculus is a fundamental aspect of calculus focusing on the concept of integration, which is essentially finding the accumulation of quantities. It is the mathematical process of finding the integral of a function.
Consider an integral in the form of \(\int_{a}^{b} f(x) \, dx\). This integral signifies the area under the curve of the function \(f(x)\) from \(a\) to \(b\). To accurately find this area, we need to understand its fundamental theorem - which connects differentiation with integration, implying how integral calculus helps reverse what is achieved through differentiation.
The process generally involves finding the antiderivative of a function and then evaluating it at the boundaries \(b\) and \(a\). For example, the integral \(\int_{1}^{e} \frac{1}{x} \, dx\) can be calculated using the antiderivative of \(\frac{1}{x}\), which is \(\ln|x|\). This results in the expression \([\ln|x|]_1^e\), simplifying to \(1\), which represents the exact value of the area under the curve from 1 to \(e\).

Error Analysis

Error analysis in numerical methods involves understanding the deviation between the approximate and exact solutions often known as the error. Specifically concerning integration methods like Simpson's Rule, error analysis helps ascertain how close our approximation is to the real value.
For the Simpson's Rule approximation, the absolute error \(\varepsilon\) can be represented as the absolute difference between the exact integral value and the approximation. This error gives us insight into how reliable our numerical method is. In this exercise, we calculated \(\varepsilon\) using the formula \(\varepsilon = |1 - \text{Simpson's Approximation}|\). This shows how much the approximation differs from the actual value \(1\) of the integral.
Furthermore, the error term for Simpson's Rule can be expressed as:

• \(\varepsilon = \frac{(b-a)^5}{180 \, N^4} \left|f^{(4)}(c)\right|\)

This utilizes the fourth derivative of the function \(f(x)\) and demonstrates how the error relates to the interval width \((b-a)\) and the number of subdivisions \(N\). Thus, it helps to understand how more subdivisions reduce error, providing a tighter approximation.

Numerical Integration

Numerical integration refers to methods for approximating the value of definite integrals. Techniques like Simpson's Rule are designed for this purpose, exploiting the fact that not all integrals can be expressed with elementary functions.
Simpson's Rule, in particular, approximates the region under a curve by dividing it into parabolic sections. Its formula is:

• \[S_N = \frac{b-a}{3N} \left( f(x_0) + 4 \sum_{i \text{ odd}} f(x_i) + 2 \sum_{i \text{ even}} f(x_i) + f(x_N) \right)\]

Here, the function is evaluated at evenly spaced points across the interval \([a, b]\), which are known as nodes. Simpson's Rule uses these nodes and their function values weighted by coefficients to compute the integral. In the given problem, with \(N=4\) and the interval \([1, e]\), we compute each function value at these nodes and apply Simpson's Rule to approximate the integral.
This approach provides a highly accurate approximation due to its higher-order error term compared to simpler numerical methods like the trapezoidal rule.

Mean Value Theorem

The Mean Value Theorem (MVT) for integrals is a critical concept in calculus that essentially states that for a continuous function on an interval \([a, b]\), there exists a point \(c\) within \((a, b)\) such that the integral over \([a, b]\) equals the function value at \(c\) times the interval length. This theorem connects the average rate of change with instantaneous rates of change.
When applied in numerical analysis and error analysis, as seen in Simpson's Rule, it yields insight into the distribution of error as related to the function's derivative and interval segment. In this exercise, the error term related to Simpson’s Rule exhibits a form that resembles the MVT:

• \(\varepsilon = \frac{(b-a)^5}{180 \, N^4} \left|f^{(4)}(c)\right|\)

This form illustrates how the fourth derivative \(f^{(4)}(c)\), evaluated at some point \(c\) in \( (a, b) \), influences the error, aligning with how MVT identifies a particular instantaneous rate that parallels average change.
Understanding this concept is crucial as it gives context to how numerical methods approximate integration results and how derivatives influence the accuracy and reliability of these approximations.