Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN: 9781285741550
Author: James Stewart
Publisher: Cengage Learning
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Riemann Sum
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Bernhard Riemann's integral was the first systematic description of the integral of a function on an interval in the branch of mathematics known as real analysis.
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### Summation and Integral Equivalence Given the summation formula: \[ L_n = \frac{4}{n} \sum_{i=1}^{n} \left[ 5 \left( 4 + (i-1) \frac{4}{n} \right)^{2} - 8 \left( 4 + (i-1) \frac{4}{n} \right)^{4} \right], \] we aim to express the limit as \( n \to \infty \) as a definite integral. Specifically, we need to determine the values of \( a \), \( b \), and \( f(x) \) in the following integral expression: \[ \int_{a}^{b} f(x) \, dx. \] ### Explanation: 1. **Interpretation of the Summation:** - Observe that the term \(\frac{4}{n}\) in the expression can be related to the width of subintervals in a Riemann sum, which is commonly used in the definition of definite integrals. - The expression inside the summation resembles a function evaluated at discrete points, which are \(4 + (i-1) \frac{4}{n}\). 2. **Connecting to Definite Integral:** - As \( n \to \infty \), the term \(\frac{4}{n}\) represents the differential element \(dx\). - The variable \(i\) runs from 1 to \(n\), which translates the function evaluation across the entire interval for integration. 3. **Determining Limits of Integration \(a\) and \(b\):** - The points \(4 + (i-1) \frac{4}{n}\) range from \(4\) to \(4 + (n-1) \frac{4}{n}\). - As \( n \to \infty \), the range approaches from \(4\) to \(8\). - Hence, the limits of integration are \(a = 4\) and \(b = 8\). 4. **Determining the Function \(f(x)\):** - The expression inside the summation can be interpreted as the function to be integrated. - The function being squared and raised to the fourth power is \(5\left(4 + (i-1) \frac{4}{
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Transcribed Image Text:### Summation and Integral Equivalence Given the summation formula: \[ L_n = \frac{4}{n} \sum_{i=1}^{n} \left[ 5 \left( 4 + (i-1) \frac{4}{n} \right)^{2} - 8 \left( 4 + (i-1) \frac{4}{n} \right)^{4} \right], \] we aim to express the limit as \( n \to \infty \) as a definite integral. Specifically, we need to determine the values of \( a \), \( b \), and \( f(x) \) in the following integral expression: \[ \int_{a}^{b} f(x) \, dx. \] ### Explanation: 1. **Interpretation of the Summation:** - Observe that the term \(\frac{4}{n}\) in the expression can be related to the width of subintervals in a Riemann sum, which is commonly used in the definition of definite integrals. - The expression inside the summation resembles a function evaluated at discrete points, which are \(4 + (i-1) \frac{4}{n}\). 2. **Connecting to Definite Integral:** - As \( n \to \infty \), the term \(\frac{4}{n}\) represents the differential element \(dx\). - The variable \(i\) runs from 1 to \(n\), which translates the function evaluation across the entire interval for integration. 3. **Determining Limits of Integration \(a\) and \(b\):** - The points \(4 + (i-1) \frac{4}{n}\) range from \(4\) to \(4 + (n-1) \frac{4}{n}\). - As \( n \to \infty \), the range approaches from \(4\) to \(8\). - Hence, the limits of integration are \(a = 4\) and \(b = 8\). 4. **Determining the Function \(f(x)\):** - The expression inside the summation can be interpreted as the function to be integrated. - The function being squared and raised to the fourth power is \(5\left(4 + (i-1) \frac{4}{
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