Problem 11 - Calculus Early Transcendentals: Pearson New International Edition

Question

Determine an \(n\) so that the Trapezoidal Rule will approximate the integral with an error \(E_{n}\) satisfying \(\left|E_{n}\right| \leq 0.01 .\) Then, using that \(n\), approximate the integral. $$ \int_{1}^{3} \frac{1}{x} d x $$

Step-by-Step Solution

Answer

The integral is approximately 1.791 using \(n = 12\).

Step 1: Calculate the Second Derivative of the Integrand

The function we are integrating is \( f(x) = \frac{1}{x} \). We need to find its second derivative.First, find the first derivative:\[f'(x) = -\frac{1}{x^2}\]Now, find the second derivative:\[f''(x) = \frac{2}{x^3}\]

Step 2: Determine the Maximum of the Second Derivative

To determine the error bound, we need the maximum of the absolute value of the second derivative over the interval \([1,3]\).The absolute value of the second derivative is:\[\left| f''(x) \right| = \frac{2}{x^3}\].This function is decreasing on \([1,3]\) so the maximum value occurs at \(x = 1\). Therefore,\[M = \frac{2}{1^3} = 2\].

Step 3: Apply the Error Bound Formula for the Trapezoidal Rule

The error bound formula for the Trapezoidal Rule is:\[\left|E_{n}\right| \leq \frac{(b-a)^{3}}{12n^{2}} M\].Substitute the known values: - \(M = 2\),- \(a = 1\),- \(b = 3\),- error \(\leq 0.01\).This becomes:\[\frac{(3-1)^{3}}{12n^{2}} \times 2 \leq 0.01\].

Step 4: Solve for \(n\)

Calculate the value of \(n\) by solving the inequality:\[\frac{8}{12n^2} \times 2 \leq 0.01\]Simplifying,\[\frac{16}{12n^2} \leq 0.01\]This can be rewritten as:\[\frac{4}{3n^2} \leq 0.01\]Multiply both sides by \(3n^2\):\[4 \leq 0.03n^2\]Divide both sides by \(0.03\):\[133.33 \leq n^2\]Take the square root of both sides:\[n \geq \sqrt{133.33} \approx 11.55\]Thus, \(n\) must be at least 12.

Step 5: Approximate the Integral Using the Trapezoidal Rule

With \(n = 12\), calculate the approximation. The formula for the Trapezoidal Rule is:\[T_n = \frac{b-a}{2n} \left[f(a) + 2 \sum_{i=1}^{n-1} f\left(a + i\frac{b-a}{n}\right) + f(b)\right]\]Substitute the values:\[T_{12} = \frac{3-1}{24} \left[\frac{1}{1} + 2 \sum_{i=1}^{11} \frac{1}{1 + \frac{2i}{12}} + \frac{1}{3}\right]\]Calculate the sum and the entire expression to approximate the integral.

Key Concepts

Error Bound

The concept of an error bound in numerical integration, such as the Trapezoidal Rule, is essential for understanding how closely an approximation aligns with the actual value of an integral. Error bounds provide a way to estimate the maximum possible error in the numerical approximation. This gives us a measure of confidence in our approximation.

When using the Trapezoidal Rule, the error bound is calculated using the formula: \[|E_n| \leq \frac{(b-a)^3}{12n^2} M\] Here,

• \((b-a)\) is the range of integration (in our exercise, it is from 1 to 3),
• \(n\) is the number of subintervals used in the approximation, and
• \(M\) is the maximum value of the absolute second derivative of the integrand over the interval.

To ensure that the approximation error is within a set tolerance, the number of subintervals \(n\) must be sufficiently large. In the exercise, we derived that with \(n = 12\), the error bound satisfies \(|E_n| \leq 0.01\). This ensures our approximation is both accurate and reliable.

Integral Approximation

Integral approximation is a method used to find the value of a definite integral when the exact solution is complex or impossible to obtain analytically. The Trapezoidal Rule is a common technique employed in such contexts.

This rule works by dividing the interval of integration into smaller subintervals. The function is approximated as a series of trapezoids rather than using a perfectly smooth function curve over each subinterval. The areas of these trapezoids are then summed to approximate the total area under the curve.

• The formula for the Trapezoidal Rule is:

\[T_n = \frac{b-a}{2n} \left[f(a) + 2 \sum_{i=1}^{n-1} f\left(a + i\frac{b-a}{n}\right) + f(b)\right]\] By plugging in the values from our exercise, we calculate this approximate integral. For \(n=12\), the approximation involves evaluating the function \(f(x) = \frac{1}{x}\) at several points and computing the sum to provide an estimate of the integral \(\int_{1}^{3} \frac{1}{x} \, dx\).

Second Derivative

The second derivative of a function plays a crucial role in determining the precision of numerical integration methods, such as the Trapezoidal Rule. It is essential in calculating the error bound because it indicates how the slope of the tangent line to a curve changes.

For the function \( f(x) = \frac{1}{x} \), the second derivative is calculated as:\[f''(x) = \frac{2}{x^3}\]The second derivative helps us assess how curved the function is over the interval of integration.

• The greater the magnitude of the second derivative, the more potential error in the approximation.
• Finding its maximum value over the interval gives \(M\), the constant used in the error bound formula.

In our exercise, the maximum is computed over the range \([1,3]\), resulting in \(M = 2\), crucial for ensuring that the error \(|E_n|\) remains within acceptable limits.

Numerical Integration

Numerical integration is a process used to approximate the value of a definite integral, especially when the integral cannot be solved analytically. Several techniques exist, such as the Trapezoidal Rule, Simpson's Rule, and others.

Choosing the right method depends on the specific function and desired accuracy. For the Trapezoidal Rule specifically, the approach assumes the area under the curve can be represented by trapezoids formed between subintervals of the function's domain.

• This method is advantageous due to its simplicity and ease of implementation.
• It provides a quick approximation that improves as the number of intervals \(n\) increases.

The process involves evaluating the function at key points, averaging adjacent points, and multiplying by the interval width. Numerical integration is particularly useful in engineering and scientific calculations where exact answers are not immediately necessary, but high precision is still required. In our exercise, applying numerical integration helps compute the integral \(\int_{1}^{3} \frac{1}{x} \, dx\) with sufficient accuracy given a specified error tolerance.