Problem 1
Question
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 (( 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 a percentage of the integral's true value. $$ \int_{1}^{2} x d x $$
Step-by-Step Solution
Verified Answer
Both Trapezoidal and Simpson's Rule show no error with a percentage error of 0%.
1Step 1: Estimate Integral using Trapezoidal Rule
To apply the Trapezoidal Rule, with \( n = 4 \) steps on the integral \( \int_{1}^{2} x \, dx \), we first calculate the step size \( h \) as follows:\[ h = \frac{b - a}{n} = \frac{2 - 1}{4} = 0.25. \]Next, evaluate the function at discrete points \( x_0 = 1, x_1 = 1.25, x_2 = 1.5, x_3 = 1.75, x_4 = 2 \):- \( f(x_0) = 1 \)- \( f(x_1) = 1.25 \)- \( f(x_2) = 1.5 \)- \( f(x_3) = 1.75 \)- \( f(x_4) = 2 \)The trapezoidal approximation is: \[ T_n = \frac{h}{2} \left[ f(x_0) + 2f(x_1) + 2f(x_2) + 2f(x_3) + f(x_4) \right] = \frac{0.25}{2} [1 + 2(1.25) + 2(1.5) + 2(1.75) + 2] = 1.5. \]
2Step 2: Calculate True Integral Value
Evaluate the integral \( \int_{1}^{2} x \, dx \) directly by calculating:\[ \left[ \frac{x^2}{2} \right]_1^2 = \frac{2^2}{2} - \frac{1^2}{2} = \frac{4}{2} - \frac{1}{2} = 1.5. \]The true value of the integral is \( 1.5 \).
3Step 3: Calculate Trapezoidal Error \( |E_T| \)
Compute the error as the absolute difference between the true value and the trapezoidal estimation, \( |E_T| = |1.5 - 1.5| = 0 \). The upper bound for \( \left|E_{T}\right| \) using the second derivative of \( f(x) = x \) (which is zero) confirms no theoretical error for constant derivatives here.
4Step 4: Trapezoidal Error Percentage
Express the trapezoidal error \( |E_T| \) as a percentage of the true integral value as:\[ \left( \frac{|E_T|}{\text{True Value}} \right) \times 100 = \left( \frac{0}{1.5} \right) \times 100 = 0\% \]
5Step 5: Estimate Integral using Simpson's Rule
With \( n = 4 \) steps, the Simpson's Rule approximation for the integral is calculated similarly:\[ S_n = \frac{h}{3} \left[ f(x_0) + 4f(x_1) + 2f(x_2) + 4f(x_3) + f(x_4) \right] = \frac{0.25}{3} [1 + 4(1.25) + 2(1.5) + 4(1.75) + 2] = 1.5 \]The approximation results in \( S_n = 1.5 \).
6Step 6: Calculate Simpson's Error \( |E_S| \)
The error for Simpson's Rule is calculated similarly,\( |E_S| = |1.5 - 1.5| = 0 \)Confirming no error occurs here as well by formula characteristics.
7Step 7: Simpson's Error Percentage
Convert \( |E_S| \) into percentage of true value:\[ \left( \frac{|E_S|}{\text{True Value}} \right) \times 100 = \left( \frac{0}{1.5} \right) \times 100 = 0\% \]
Key Concepts
Trapezoidal RuleSimpson's RuleIntegration Error Analysis
Trapezoidal Rule
The Trapezoidal Rule is a simple technique used in numerical integration to approximate the definite integral of a function. It works by approximating the region under the function's graph, between two points, using trapezoids. For the integral of our exercise, \( \int_{1}^{2} x \, dx \), the Trapezoidal Rule calculates an approximate value by dividing the interval \([1, 2]\) into \(n\) equally spaced subintervals. In this case, \(n = 4\) resulting in a step size \(h = 0.25\).
The function is evaluated at discrete points \(x_0, x_1, x_2, x_3, x_4\) where values are \(x_0 = 1\), \(x_1 = 1.25\), etc. The trapezoid approximation then uses these values in the formula:
The error piece \(|E_T|\) in trapezoid approximation generally represents the difference between the approximate and actual integral values. For linear functions like \(f(x) = x\), the error can be zero because the curvature that usually causes trapezoid approximation error is absent.
The function is evaluated at discrete points \(x_0, x_1, x_2, x_3, x_4\) where values are \(x_0 = 1\), \(x_1 = 1.25\), etc. The trapezoid approximation then uses these values in the formula:
- \(T_n = \frac{h}{2} \left[ f(x_0) + 2f(x_1) + 2f(x_2) + 2f(x_3) + f(x_4) \right]\)
The error piece \(|E_T|\) in trapezoid approximation generally represents the difference between the approximate and actual integral values. For linear functions like \(f(x) = x\), the error can be zero because the curvature that usually causes trapezoid approximation error is absent.
Simpson's Rule
Simpson's Rule is another method for numerical integration, typically more accurate than the Trapezoidal Rule. It fits a parabola under the curve instead of a straight line. The method is ideal for functions that are at least quadratic, providing a very close approximation with fewer subintervals. Here, we apply Simpson's Rule on the integral \(\int_{1}^{2} x \, dx\) using \(n=4\) subintervals.
Just like with the Trapezoidal Rule, the function is evaluated at discrete points: \(x_0, x_1, ... , x_4\). Simpson's approximation formula is:
Simpson's Rule usually results in smaller errors because the approximation follows the function's curvature more closely. However, for simple functions where the true integral is already captured perfectly, like \(f(x) = x\) here, the error \(|E_S|\) is zero, showing no loss of precision.
Just like with the Trapezoidal Rule, the function is evaluated at discrete points: \(x_0, x_1, ... , x_4\). Simpson's approximation formula is:
- \(S_n = \frac{h}{3} \left[ f(x_0) + 4f(x_1) + 2f(x_2) + 4f(x_3) + f(x_4) \right]\)
Simpson's Rule usually results in smaller errors because the approximation follows the function's curvature more closely. However, for simple functions where the true integral is already captured perfectly, like \(f(x) = x\) here, the error \(|E_S|\) is zero, showing no loss of precision.
Integration Error Analysis
Integrating functions numerically yields approximations instead of exact values. Therefore, understanding error analysis is crucial. This process involves estimating the difference between the calculated approximation and the actual integral.
With the Trapezoidal Rule, the error \( |E_T| \) is typically influenced by how well straight line segments approximate the curve. This error can be reduced by increasing \(n\), the number of subintervals, or intrinsically by the simplicity of the function such as linear cases or when higher derivatives lead to zero. This was the case in this problem, where \( |E_T| = 0 \).
For Simpson's Rule, the error \( |E_S| \) often tends to be even smaller than that of the Trapezoidal Rule. The method employs second-order polynomials to fit curves, capturing more complex variations in the function. In this exercise, the error is again \(0\), which confirms the method's capabilities or simply the linear nature of the function. When errors are expressed as percentages, both methods here result in \(0\%\) of the true value, signifying perfect approximation in this case.
Error analysis is key to deciding between methods and understanding their limitations or benefits in various contexts of function integration.
With the Trapezoidal Rule, the error \( |E_T| \) is typically influenced by how well straight line segments approximate the curve. This error can be reduced by increasing \(n\), the number of subintervals, or intrinsically by the simplicity of the function such as linear cases or when higher derivatives lead to zero. This was the case in this problem, where \( |E_T| = 0 \).
For Simpson's Rule, the error \( |E_S| \) often tends to be even smaller than that of the Trapezoidal Rule. The method employs second-order polynomials to fit curves, capturing more complex variations in the function. In this exercise, the error is again \(0\), which confirms the method's capabilities or simply the linear nature of the function. When errors are expressed as percentages, both methods here result in \(0\%\) of the true value, signifying perfect approximation in this case.
Error analysis is key to deciding between methods and understanding their limitations or benefits in various contexts of function integration.
Other exercises in this chapter
Problem 1
Use the table of integrals at the back of the book to evaluate the integrals. \(\int \frac{d x}{x \sqrt{x-3}}\)
View solution Problem 1
Expand the quotients in Exercises \(1-8\) by partial fractions. $$\frac{5 x-13}{(x-3)(x-2)}$$
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Evaluate the integrals. $$\int \frac{d x}{\sqrt{9+x^{2}}}$$
View solution Problem 1
Evaluate the integrals in Exercises \(1-22\) $$ \int \cos 2 x d x $$
View solution