91Ó°ÊÓ

The instructions for the integrals in Exercises \(1-10\) have two parts, one for the Trapezoidal Rule and one for Simpson's Rule. I. Using the Trapezoidal Rule a. Estimate the integral with \(n=4\) steps and find an upper bound for \(\left|E_{T}\right| .\) b. Evaluate the integral directly and find \(\left|E_{T}\right|\) c. Use the formula \((\left | E_{T}\right | /(\) true value)) \(\times 100\) to express \(\left|E_{T}\right|\) as \right.\right. a percentage of the integral's true value. II. Using Simpson's Rule a. Estimate the integral with \(n=4\) steps and find an upper bound for \(\left|E_{S}\right| .\) b. Evaluate the integral directly and find \(\left|E_{S}\right|\) c. Use the formula \((\left|E_{S}\right| /(\) true value) \() \times 100\) to express \(\left|E_{S}\right|\) as \right. \(\quad\) a percentage of the integral's true value. The instructions for the integrals in Exercises \(1-10\) have two parts, one for the Trapezoidal Rule and one for Simpson's Rule. I. Using the Trapezoidal Rule a. Estimate the integral with \(n=4\) steps and find an upper bound for \(\left|E_{T}\right| .\) b. Evaluate the integral directly and find \(\left|E_{T}\right|\) c. Use the formula \((\left | E_{T}\right | /(\) true value)) \(\times 100\) to express \(\left|E_{T}\right|\) as \right.\right. a percentage of the integral's true value. II. Using Simpson's Rule a. Estimate the integral with \(n=4\) steps and find an upper bound for \(\left|E_{S}\right| .\) b. Evaluate the integral directly and find \(\left|E_{S}\right|\) c. Use the formula \((\left|E_{S}\right| /(\) true value) \() \times 100\) to express \(\left|E_{S}\right|\) as \right. \(\quad\) a percentage of the integral's true value. The instructions for the integrals in Exercises \(1-10\) have two parts, one for the Trapezoidal Rule and one for Simpson's Rule. I. Using the Trapezoidal Rule a. Estimate the integral with \(n=4\) steps and find an upper bound for \(\left|E_{T}\right| .\) b. Evaluate the integral directly and find \(\left|E_{T}\right|\) c. Use the formula \((\left | E_{T}\right | /(\) true value)) \(\times 100\) to express \(\left|E_{T}\right|\) as \right.\right. a percentage of the integral's true value. II. Using Simpson's Rule a. Estimate the integral with \(n=4\) steps and find an upper bound for \(\left|E_{S}\right| .\) b. Evaluate the integral directly and find \(\left|E_{S}\right|\) c. Use the formula \((\left|E_{S}\right| /(\) true value) \() \times 100\) to express \(\left|E_{S}\right|\) as \right. \(\quad\) a percentage of the integral's true value. $$ \int_{0}^{1} \sin \pi t d t $$

Step by step solution

01

Calculate the Step Size, h

For the interval \([0, 1]\) and \(n = 4\) steps, the step size \(h\) is calculated as \(h = \frac{b-a}{n} = \frac{1-0}{4} = 0.25\).

02

Apply the Trapezoidal Rule Formula

The Trapezoidal Rule formula is \(T_n = \frac{h}{2} [f(x_0) + 2(f(x_1) + f(x_2) + ... + f(x_{n-1})) + f(x_n)]\). The subintervals are \(x_0 = 0\), \(x_1 = 0.25\), \(x_2 = 0.5\), \(x_3 = 0.75\), \(x_4 = 1\). Now, calculate each function value:- \(f(0) = \sin\pi \cdot 0 = 0\),- \(f(0.25) = \sin\pi \cdot 0.25 = 1\),- \(f(0.5) = \sin\pi \cdot 0.5 = 0\),- \(f(0.75) = \sin\pi \cdot 0.75 = -1\),- \(f(1) = \sin\pi \cdot 1 = 0\).Thus, \(T_4 = \frac{0.25}{2} [0 + 2(1 + 0 - 1) + 0] = 0\).

03

Calculate the True Value of the Integral

Evaluate the integral directly: \(\int_{0}^{1} \sin \pi t\, dt = -\frac{1}{\pi} [ \cos \pi t]_{0}^{1} = -\frac{1}{\pi} (\cos \pi \cdot 1 - \cos \pi \cdot 0) = \frac{2}{\pi}\).

04

Calculate the Trapezoidal Rule Error Term, E_T

The error term formula for the Trapezoidal Rule is \(\left|E_T\right| = \frac{K(b-a)^3}{12n^2}\). The second derivative of \(\sin \pi t\) is \(-\pi^2 \sin \pi t\), which reaches its maximum of \(\pi^2\) in the given interval. Therefore, \(E_T = \frac{\pi^2(1)^3}{12 \cdot 4^2} = \frac{\pi^2}{192}\).

05

Convert Trapezoidal Error to Percentage

The percentage error is calculated using the formula \(\left(\left|E_T\right| / \text{true value}\right) \times 100\). Therefore, this is \(\left(\frac{\pi^2 / 192}{2/\pi}\right) \times 100\approx 2.57\%\).

06

Apply Simpson's Rule Formula

Simpson's Rule formula is \(S_n = \frac{h}{3} [f(x_0) + 4f(x_1) + 2f(x_2) + 4f(x_3) + f(x_4)]\). Using the subintervals: - \(S_4 = \frac{0.25}{3} [0 + 4 \times 1 + 2 \times 0 + 4 \times (-1) + 0] = 0\).

07

Calculate the Simpson's Rule Error Term, E_S

The error term formula for Simpson’s Rule is \(\left|E_S\right| = \frac{K(b-a)^5}{180n^4}\). The fourth derivative of \(\sin \pi t\) is \(-\pi^4 \sin \pi t\), which reaches its maximum of \(\pi^4\). Therefore, \(E_S = \frac{\pi^4(1)^5}{180 \cdot 4^4} = \frac{\pi^4}{46080}\).

08

Convert Simpson's Error to Percentage

The percentage error is calculated using the formula \(\left(\left|E_S\right| / \text{true value}\right) \times 100\). Therefore, this is \(\left(\frac{\pi^4 / 46080}{2/\pi}\right) \times 100\approx 0.042\%\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Trapezoidal Rule

The Trapezoidal Rule is a fundamental method for estimating the value of a definite integral. It is particularly useful when the analytical solution of the integral is not easily attainable. This rule approximates the region under the curve as a series of trapezoids and then calculates their collective area.
The basic formula is given by:\[ T_n = \frac{h}{2} \left[f(x_0) + 2(f(x_1) + f(x_2) + \dots + f(x_{n-1})) + f(x_n) \right] \]where:

• \(h\) is the step size, calculated as \((b-a)/n\),
• \(n\) denotes the number of trapezoids,
• \(f(x_k)\) are the function values at the endpoints and midpoints.

Using the given example with \(n=4\) over the interval \([0, 1]\), the midpoint values need to be calculated.
This helps break down complex integrals into easier-to-solve components.

Simpson's Rule

Simpson's Rule is another powerful tool for numerical integration. This approach uses parabolic segments instead of straight lines for more accurate approximation compared to the Trapezoidal Rule. It's perfect when you need a higher degree of precision with fewer intervals.
The formula for Simpson's Rule is:\[ S_n = \frac{h}{3} \left[f(x_0) + 4(f(x_1) + f(x_3) + \cdots ) + 2(f(x_2) + f(x_4) + \cdots ) + f(x_n) \right] \]This involves:

• Intervals that must be even,
• A combination of weights: 1 for the endpoints, 4 for odd indices, and 2 for even indices between them.

In our example, Simpson's Rule gives a more precise approximation with fewer intervals. It makes it ideal for functions resembling curves more than lines.

Integral Approximation

Integral approximation is the process of finding the numerical value of a definite integral. When the exact analytical evaluation of an integral is challenging, approximation methods like the Trapezoidal Rule and Simpson's Rule become invaluable. They provide close estimates to the true integral value
, bridging the gap between complex calculus and solvable numerical problems.
In practical applications:

• These rules are often implemented in computational software,
• They enable the handling of real-world data and models.

By understanding their formulation and implementation, one can select the most appropriate technique based on the required accuracy and computational efficiency.

Error Analysis

Error analysis in numerical integration helps us understand how close or far our approximation is from the true value. For both the Trapezoidal and Simpson's Rules, knowing the error involved is crucial for assessing the reliability of the approximation.
For the Trapezoidal Rule, the error term is:\[ \left|E_T\right| = \frac{K(b-a)^3}{12n^2} \]where \(K\) is the maximum of the second derivative in the interval.For Simpson's Rule, it is:\[ \left|E_S\right| = \frac{K(b-a)^5}{180n^4} \]where \(K\) includes the maximum of the fourth derivative.Calculating these errors allows adjusting methods to reduce inaccuracies:

• Increasing the number of intervals \(n\),
• Estimating the applicable \(K\) precisely for better bounds.

By converting errors to percentages, we place their impact into context, making complex data easier to understand and apply.