Problem 65 - 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 \((5.8 .4) .)\) $$ \int_{1}^{4} \sqrt{x} d x \quad N=6 $$

Step-by-Step Solution

Answer

The exact integral is \( \frac{14}{3} \), Simpson's approximation \( S_6 \approx 4.734 \), and error \( \varepsilon \approx 0.067 \).

Step 1: Compute the Exact Value of the Integral

The integral \( \int_{1}^{4} \sqrt{x} \, dx \) is a definite integral of the function \( f(x) = \sqrt{x} \). We begin by finding the antiderivative of \( \sqrt{x} \), which is \( \frac{2}{3} x^{3/2} \). Applying the limits from 1 to 4, we have: \[ \int_{1}^{4} \sqrt{x} \, dx = \left[ \frac{2}{3} x^{3/2} \right]_{1}^{4} = \frac{2}{3} (4)^{3/2} - \frac{2}{3} (1)^{3/2} \] Calculating further, we find \( 4^{3/2} = 8 \) and \( 1^{3/2} = 1 \), thus: \[ \frac{2}{3} \cdot 8 - \frac{2}{3} \cdot 1 = \frac{16}{3} - \frac{2}{3} = \frac{14}{3} \]\.

Step 2: Apply Simpson's Rule Approximation

Simpson's Rule for approximating the integral over \([a, b]\) with \( N = 6 \) subintervals is given by: \[ S_N = \frac{\Delta x}{3} \left( f(x_0) + 4 \sum_{k=1, 3, 5} f(x_k) + 2 \sum_{k=2, 4} f(x_k) + f(x_6) \right) \] Where \( \Delta x = \frac{b-a}{N} = \frac{3}{6} = 0.5 \). The nodes are \( x_0 = 1, x_1 = 1.5, \ldots, x_6 = 4 \). Calculate \( f(x_k) = \sqrt{x_k} \) at each node and sum as per Simpson's coefficients: \( S_6 = \frac{0.5}{3} \left( \sqrt{1} + 4\sqrt{1.5} + 2\sqrt{2} + 4\sqrt{2.5} + 2\sqrt{3} + 4\sqrt{3.5} + \sqrt{4} \right) \). Calculating gives \( S_6 \approx 4.734 \).

Step 3: Calculate the Absolute Error

The absolute error \( \varepsilon \) is given by the difference between the exact value and the Simpson's approximation: \[ \varepsilon = \left| \frac{14}{3} - 4.734 \right| = \left| 4.6667 - 4.734 \right| = 0.0673 \approx 0.067 \].

Step 4: Finding the Value of \( c \)

According to the error formula, \[ \varepsilon = \frac{(b-a)^{5} \left| f^{(4)}(c) \right|}{180 \cdot N^{4}} \] Substitute \( a = 1 \), \( b = 4 \), and \( N = 6 \), \[ 0.067 = \frac{3^5 \left| f^{(4)}(c) \right|}{180 \times 6^4} \] Simplify: \[ 0.067 = \frac{243 \times |f^{(4)}(c)|}{233280} \] \[ |f^{(4)}(c)| \approx 6.423 \text{ (solve to match the error)} \]. Find an appropriate \( c \) in the interval \( (1, 4) \) from the function \( f^{(4)}(x) \).

Key Concepts

Integral Approximation

Simpson's Rule is a powerful technique used for integral approximation, particularly when the function in question is difficult to integrate analytically. It is a method that estimates the definite integral by approximating the region under the curve of a function using a series of parabolic arcs instead of rectangles (as in the Riemann sums). This method is particularly effective when the interval is subdivided into an even number of segments, or subintervals.

• The main idea is to take the integral over small subintervals and approximate the area under the curve of each segment with a parabola.
• The more subintervals we use, the smaller the error in our approximation, as the parabolic curves adhere more closely to the actual curvature of the function.
• Simpson's Rule is generally expressed for an integral from a to b as:\[ S_N = \frac{\Delta x}{3} \left( f(x_0) + 4 \sum_{k=1, 3, 0…} f(x_k) + 2 \sum_{k=2, 4, 4…} f(x_k) + f(x_N) \right) \]where \Delta x = \frac{b-a}{N} and N is the number of subintervals.

Using Simpson's Rule for the given integral \( \int_{1}^{4} \sqrt{x} \, dx \) with N=6, we obtain a close approximation to the exact value. Calculations involve evaluating the function \( \sqrt{x} \) at specific points within the interval and then applying them within the Simpson's Rule formula.

Absolute Error Calculation

The absolute error in numerical methods like Simpson's Rule measures the difference between the approximate and exact values of the integral. It is an important step in assessing the accuracy of the approximation method.

• In general, absolute error \( \varepsilon \) is defined as the absolute value of the difference between the exact value of the integral and its approximate value obtained via Simpson's Rule:
\[ \varepsilon = | I_{ ext{exact}} - I_{ ext{approx}} | \]
• For our specific example, the exact integral value of \( \int_{1}^{4} \sqrt{x} dx \) was calculated to be approximately \( \frac{14}{3} \approx 4.667 \), and the Simpson's approximation gave us \( S_6 \approx 4.734 \).
• The absolute error is then \( \varepsilon = | 4.667 - 4.734 | = 0.067 \).

Understanding this error gives students insight into how reliable the approximation is when comparing to the true, analytical integral value.

Error Formula

The error formula for Simpson's Rule is crucial in understanding the potential deviation from the exact value and involves higher derivatives of the function being integrated. The formula states that:\[\varepsilon = \frac{(b-a)^5 \left| f^{(4)}(c) \right|}{180 \cdot N^4}\]where \( f^{(4)}(c) \) is the fourth derivative of the function evaluated at some point c in the interval (a, b). This is derived from a bound on the error and closely resembles the Mean Value Theorem for integrals.

• This formula helps in quantifying the maximum error bound that Simpson's Rule might introduce for the chosen interval size and number of partitions.
• Finding \( c \) such that this equation holds involves ensuring consistency with the calculated absolute error. This often requires solving for \( c \) by trial or understanding behavior of the function's higher derivatives within the interval.

In this problem, we set the calculated absolute error equal to this error expression to find \( |f^{(4)}(c)| \approx 6.423 \), providing insight into what the curvature of \( \sqrt{x} \) might allow for precise bounds.

Definite Integral Calculation

Definite integrals provide the signed area under a curve for a given function over a specified interval. In this problem, we tackled the definite integral \( \int_{1}^{4} \sqrt{x} \, dx \), which involves solving the problem using calculus techniques.

• Calculating a definite integral initially involves finding the antiderivative of the function being integrated, which is the inverse process of differentiation.
• For \( f(x) = \sqrt{x} \), the antiderivative is \( \frac{2}{3} x^{3/2} \).
• Once the antiderivative is determined, we apply the limits of integration to compute the total area, i.e., \( \int_{1}^{4} \sqrt{x} dx = \left[ \frac{2}{3} x^{3/2} \right]_1^4 = \frac{16}{3} - \frac{2}{3} = \frac{14}{3} \).

This exact calculation gives a true representation compared to approximation, and offers a benchmark against which numerical estimates like those from Simpson's Rule can be measured. Understanding the principles of tackling definite integrals is foundational in analysis and applications across mathematics and the sciences.