Problem 12 - Thomas Calculus

Question

In Exercises \(11-14\) , use the tabulated values of the integrand to estimate the integral with (a) the Trapezoidal Rule and (b) Simpson's Rule with \(n=8\) steps. Round your answers to five decimal places. Then (c) find the integral's exact value and the approximation error \(E_{T}\) or \(E_{s}\) as appropriate. $$ \int_{0}^{3} \frac{\theta}{\sqrt{16+\theta^{2}}} d \theta $$ $$ \begin{array}{ll}{\boldsymbol{\theta}} & {\boldsymbol{\theta} / \sqrt{16+\boldsymbol{\theta}^{2}}} \\ {0} & {0.0} \\ {0.375} & {0.09334} \\\ {0.75} & {0.18429} \\ {1.125} & {0.27075} \\ {1.125} & {0.35112} \\ {1.575} & {0.42443} \\ {2.625} & {0.49026} \\ {2.625} & {0.58466} \\ {3.0} & {0.6}\end{array} $$

Step-by-Step Solution

Answer

Trapezoidal: 1.89363, Simpson's: 1.89358, Exact: 1.89549, Errors: \(E_{T} = 0.00186\), \(E_{S} = 0.00191\).

Step 1: Define the Trapezoidal Rule Formula

The Trapezoidal Rule estimates an integral by approximating the region under the graph of the function as a series of trapezoids. The formula is: \[T = \frac{h}{2} \left( f(x_0) + 2 \sum_{i=1}^{n-1} f(x_i) + f(x_n) \right),\] where \(h\) is the step size \(\frac{b-a}{n}\), and \(f(x_i)\) are function values at specified points.

Step 2: Apply the Trapezoidal Rule

For \(n = 8\), calculate the step size \( h = \frac{3-0}{8} = 0.375\). Then apply the formula: \[T = \frac{0.375}{2} \left( 0.0 + 2(0.09334 + 0.18429 + 0.27075 + 0.35112 + 0.42443 + 0.49026 + 0.58466) + 0.6 \right).\] After calculations, \( T \approx 1.89363 \).

Step 3: Define Simpson's Rule Formula

Simpson's Rule estimates an integral by approximating the region under the graph of the function with parabolic arcs. The formula is: \[S = \frac{h}{3} \left( f(x_0) + 4 \sum_{i=1,3,5,\ldots}^{n-1} f(x_i) + 2 \sum_{i=2,4,6,\ldots}^{n-2} f(x_i) + f(x_n) \right).\]

Step 4: Apply Simpson's Rule

Apply the Simpson's Rule formula, with \(n=8\) and \(h = 0.375\): \[S = \frac{0.375}{3} \left( 0.0 + 4(0.09334 + 0.27075 + 0.42443 + 0.58466) + 2(0.18429 + 0.35112 + 0.49026) + 0.6 \right).\] After performing the calculations, \( S \approx 1.89358 \).

Step 5: Compute the Exact Integral Value

The integral \( \int_{0}^{3} \frac{\theta}{\sqrt{16+\theta^{2}}} \, d\theta \) can be solved exactly using substitution or a known formula. The exact value is \( 1.89549 \).

Step 6: Calculate the Errors

Calculate the error for Trapezoidal Rule: \( E_{T} = |1.89549 - 1.89363| \approx 0.00186 \). Calculate the error for Simpson's Rule: \( E_{S} = |1.89549 - 1.89358| \approx 0.00191 \).

Key Concepts

Trapezoidal Rule

The Trapezoidal Rule is a numerical method used to estimate the definite integral of a function. The name comes from dividing the area under a curve into a series of trapezoids and summing their areas. This method is quite useful in cases where it is difficult to find an exact integral or when dealing with tabulated data.

To apply the Trapezoidal Rule, you divide the interval of integration into smaller sub-intervals of equal length. The step size, denoted as \( h \), is given by \( h = \frac{b-a}{n} \) where \( b \) and \( a \) are the upper and lower limits of integration respectively, and \( n \) is the number of sub-intervals. With values of the function at specific points, \( f(x_0), f(x_1), \, \ldots, \, f(x_n) \), the formula for the Trapezoidal Rule is:

• \( T = \frac{h}{2} \left( f(x_0) + 2 \sum_{i=1}^{n-1} f(x_i) + f(x_n) \right) \)

In essence, the Trapezoidal Rule linearly interpolates the data, approximating the curve with line segments. While it's an easy and quick method, it may not be as precise when the function is not approximately linear. In our example, this rule resulted in an estimate of \( T \approx 1.89363 \).

Simpson's Rule

Simpson's Rule is another famous technique for numerical integration, offering potentially higher accuracy than the Trapezoidal Rule. It achieves this precision by using parabolic segments to approximate the function's path between known data points, rather than straight-line segments. This works particularly effectively when the function can be reasonably approximated by a quadratic in each interval.

Like with the Trapezoidal Rule, divide the integration range into an even number of sub-intervals. The step size remains \( h = \frac{b-a}{n} \). Then apply Simpson's Rule through this formula:

• \( S = \frac{h}{3} \left( f(x_0) + 4 \sum_{i=1,3,5,\ldots}^{n-1} f(x_i) + 2 \sum_{i=2,4,6,\ldots}^{n-2} f(x_i) + f(x_n) \right) \)

The pattern of coefficients \(4, 2,\) and \(1\) reflects the method’s quadratic nature, with the outermost points weighted once, while alternate sequential points within the interval have varying weights.

In our example, applying Simpson’s Rule resulted in an estimate \( S \approx 1.89358 \). This approach confidently tackles many functions providing improved accuracy over the Trapezoidal Rule.

Approximation Error

Approximation error is a crucial concept when working with numerical methods like the Trapezoidal and Simpson’s Rules. It measures the difference between the estimated integral value and the actual or exact integral value. Understanding this error helps in evaluating the accuracy and efficacy of different numerical approximation methods.

Calculating the approximation error involves comparing the estimated results with the exact value. For the Trapezoidal Rule, this error is given by:

• \( E_T = |V_{exact} - T| \)

For Simpson's Rule, the error is calculated similarly:

• \( E_S = |V_{exact} - S| \)

In the example provided, the exact integral value of \( \int_{0}^{3} \frac{\theta}{\sqrt{16+\theta^2}} \, d\theta \) is \( 1.89549 \). The approximation error for the Trapezoidal Rule was found to be \( E_T \approx 0.00186 \), indicating that the estimated value was slightly below the actual integral. However, with Simpson's Rule, \( E_S \approx 0.00191 \), a similar order of magnitude error, emphasizing its slightly different approximation characteristics. These errors highlight the balance between computational simplicity and accuracy, enabling informed decisions regarding method selection.