Problem 19 - Thomas Calculus

Question

Use series to approximate the values of the integrals in Exercises \(19-\) 22 with an error of magnitude less than \(10^{-8} .\) \begin{equation} \int_{0}^{0.1} \frac{\sin x}{x} d x \end{equation}

Step-by-Step Solution

Answer

The approximate value of the integral is computed using a series expansion until the error is less than \(10^{-8}\).

Step 1: Express the Function as a Series

The integrand \( \frac{\sin x}{x} \) can be expanded using the Maclaurin series for \( \sin x \), which is \( \sin x = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \cdots \). Therefore, \( \frac{\sin x}{x} = 1 - \frac{x^2}{3!} + \frac{x^4}{5!} - \cdots \).

Step 2: Integrate Term by Term

Integrate the series term by term from 0 to 0.1: \[ \int_{0}^{0.1} \left(1 - \frac{x^2}{3!} + \frac{x^4}{5!} - \cdots \right) dx = \left[ x - \frac{x^3}{3! \times 3} + \frac{x^5}{5! \times 5} - \cdots \right]_{0}^{0.1}. \]

Step 3: Evaluate the Expressions

Calculate the resulting expression for each term at the upper limit 0.1 and subtract the value at the lower limit 0. For example, the first few terms are: \[ 0.1 - \frac{0.1^3}{3! \times 3} + \frac{0.1^5}{5! \times 5} - \cdots \].

Step 4: Determine the Number of Terms Needed

Calculate the value of the integral using the series until the absolute value of the last term in the series is less than \( 10^{-8} \). Continue adding terms until the desired accuracy is reached.

Step 5: Sum the Series for the Final Value

Compute the sum of the series with the necessary number of terms from Step 4. This final sum will be the approximate value of the integral.

Key Concepts

Maclaurin Series

The Maclaurin series is a special case of the Taylor series. It is used to represent functions as infinite sums of terms calculated from the values of their derivatives at a single point, which is zero for the Maclaurin series. This series can provide a powerful way to approximate functions, which is especially useful when dealing with integrals or complex functions. For the function \(\sin x\), the Maclaurin series expansion is given by: \[\sin x = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \cdots\] This expression can be divided by \(x\) to approximate \(\frac{\sin x}{x}\), leading to: \[\frac{\sin x}{x} = 1 - \frac{x^2}{3!} + \frac{x^4}{5!} - \cdots\]

• Understanding these terms: Each term of the series alternates in sign and decreases in absolute value because of the factorial in the denominator.
• Convergence: The smaller the value of \(x\), the better the series will approximate the function.

Term-by-Term Integration

Once we have our series expansion, the next step is term-by-term integration. This method involves integrating each term of the series separately, a procedure that is allowed when the series is convergent. For the function \(\frac{\sin x}{x}\), the series \(1 - \frac{x^2}{3!} + \frac{x^4}{5!} - \cdots\) is integrated from 0 to 0.1, leading to: \[\int_{0}^{0.1} \left(1 - \frac{x^2}{3!} + \frac{x^4}{5!} - \cdots\right) dx = \left[ x - \frac{x^3}{3! \times 3} + \frac{x^5}{5! \times 5} - \cdots \right]_{0}^{0.1}\]

• Simplifying the Integration: Each power of \(x\) results in an increment in the power of the integral term and division by this new power.
• Basic Integration: This step-by-step integration reflects the basic rules of calculus applied to each term.

After integration, apply the limits of 0 and 0.1 to the resulting expressions, which will help in finalizing the approximation.

Error Estimation

Ensuring that our approximation is accurate involves understanding and estimating the error. The error of a series approximation can typically be figured out by looking at the magnitude of the next unused term in the series. In this exercise, we continue adding terms until the absolute value of the last term used in our series is less than \(10^{-8}\). This threshold is set to ensure the error in our calculation is less than the desired precision.

• Importance of Error Estimation: It assures that our approximation is within the specified precision bounds.
• Practical Error Checking: In practical terms, we keep adding terms until the next term is smaller than this threshold.
• Efficiency: By stopping once this condition is met, it avoids unnecessary computation while maintaining accuracy.

This careful control over the approximation error helps guarantee that our final result is as close to the true value as required by the problem's constraints.