Problem 33 - Thomas Calculus

Question

Usable values of the sine-integral function The sine-integral function, $$ \operatorname{Si}(x)=\int_{0}^{x} \frac{\sin t}{t} d t $$ is one of the many functions in engineering whose formulas cannot be simplified. There is no elementary formula for the antiderivative of \((\sin 1) / t\) . The values of \(\mathrm{Si}(x),\) however, are readily estimated by numerical integration. Although the notation does not show it explicitly, the function being integrated is $$ f(t)=\left\\{\begin{array}{cl}{\frac{\sin t}{t},} & {t \neq 0} \\ {1,} & {t=0}\end{array}\right. $$ the continuous extension of \((\sin t) / t\) to the interval \([0, x] .\) The function has derivatives of all orders at every point of its domain. Its graph is smooth, and you can expect good results from Simpson's Rule. a. Use the fact that \(\left|f^{(4)}\right| \leq 1\) on \([0, \pi / 2]\) to give an upper bound for the error that will occur if $$ \operatorname{Si}\left(\frac{\pi}{2}\right)=\int_{0}^{\pi / 2} \frac{\sin t}{t} d t $$ is estimated by Simpson's Rule with \(n=4\) . b. Estimate \(\operatorname{Si}(\pi / 2)\) by Simpson's Rule with \(n=4\) c. Express the error bound you found in part (a) as a percentage of the value you found in part (b).

Step-by-Step Solution

Answer

Use Simpson's Rule to calculate \(\operatorname{Si}(\pi/2)\) and approximate the error as a percentage following the steps.

Step 1: Understanding Simpson's Rule and Error

Simpson's Rule is a numerical integration technique that approximates definite integrals using parabolic segments. The error for Simpson's Rule is given by \ \( E_n = -\frac{(b-a)^5}{180n^4}f^{(4)}(\xi) \) where \( \xi \) is a number in the interval \( [a, b] \). The error bound presumes \( |f^{(4)}(t)| \leq M \) on \([a,b]\). In this problem, \(f^{(4)}\) is bounded by 1.

Step 2: Calculate Error Bound in Part (a)

Given \([a, b] = [0, \pi/2]\) and \(n = 4\), we can calculate the upper bound for the error. Substitute \(a = 0\), \(b = \frac{\pi}{2}\), \(n = 4\), and \(M = 1\) into the error formula. This produces: \[ E_n = -\frac{(\pi/2 - 0)^5}{180 \cdot 4^4} \cdot 1 = -\frac{(\pi/2)^5}{46080} \approx -0.00002751 \]

Step 3: Estimate \(\operatorname{Si}(\pi/2)\) with Simpson's Rule

Simpson's Rule approximation is given by:\[ \int_{a}^{b} f(t) \, dt \approx \frac{b-a}{3n} \left( f(a) + 4 \sum \text{odd indices} f(t_i) + 2 \sum \text{even indices} f(t_i) + f(b) \right) \] The step size \( h = \frac{\pi/2 - 0}{4} = \frac{\pi}{8} \). Calculate \( f(t) \) at appropriate sample points and substitute these into Simpson's Rule formula to approximate the integral.

Step 4: Express Error Bound as a Percentage

The error percentage is calculated by dividing the error bound from part (a) by the value estimated in part (b), then multiplying by 100. \[ \text{Percentage error} = \left| \frac{E_n}{\text{Simpson Approximation}} \right| \times 100 \]Use the calculated error approximation value and the Simpson approximation value to find this percentage.

Key Concepts

Simpson's Rule

Simpson's Rule is a powerful method for approximating the integral of a function over a closed interval. Unlike other numerical methods like the Trapezoidal Rule, Simpson's Rule uses parabolic arcs instead of straight lines to approximate the function. This makes it especially useful for integrals where the function is smooth and has a continuous curvature.
One of the benefits of Simpson's Rule is its accuracy for relatively few subintervals. This is because it uses information about the function at both the ends and midpoints of the subintervals to construct quadratic polynomials that better match the actual curve. The rule is usually applied by dividing the interval \(a, b\) into an even number of subintervals \(n\) and calculating:\[ \int_{a}^{b} f(t) \, dt \approx \frac{b-a}{3n} \left( f(a) + 4 \sum \text{odd indices} f(t_i) + 2 \sum \text{even indices} f(t_i) + f(b) \right) \]
In this formula, \(t_i\) are the sample points, and the odd/even index structure brings about the weighting that gives Simpson's Rule its accuracy. The approximation method is particularly well-suited for functions resembling parabolic sections, offering reliable estimates even for functions without simple antiderivatives.

Error Bound

When using numerical methods like Simpson's Rule, it's important to understand the potential error involved. The error bound gives an upper limit on how far off the numerical estimate might be. For Simpson's Rule, this error is expressed mathematically as:
\[ E_n = -\frac{(b-a)^5}{180n^4}f^{(4)}(\xi) \]
where \(\xi\) is some value within the interval \[a, b\]. The term \(f^{(4)}(\xi)\) represents the fourth derivative of the function, indicating that the rule's error depends on the change in this derivative.
To apply this in practice, we often use a known upper bound for \( \left| f^{(4)} \right| \) over the interval to find the greatest possible error. By understanding the characteristics of the fourth derivative, one can adjust the number of subintervals \((n)\) to ensure the error stays within an acceptable range, thus making the method both practical and reliable for engineering problems.

Sine-Integral Function

The sine-integral function \(\operatorname{Si}(x)\) is used in various engineering and physics applications. Defined by the integral:
\[ \operatorname{Si}(x)=\int_{0}^{x} \frac{\sin t}{t} \, dt \]
, it presents a unique challenge because its integral does not have a simple antiderivative solution. Yet, it's an important function for certain calculations, especially in signal processing and wave analysis.
Because of these properties, engineers often turn to numerical techniques like Simpson's Rule to estimate values of \(\operatorname{Si}(x)\). By treating the integrand as \(f(t)=\frac{\sin t}{t}\) and knowing that it's smooth over intervals such as \[0, \pi/2\], accurate approximations for \(\operatorname{Si}(x)\) can be determined easily. This smoothness also means that the error will be relatively low, making numerical estimates reliable for practical use.