Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
Related questions
Concept explainers
Riemann Sum
Riemann Sums is a special type of approximation of the area under a curve by dividing it into multiple simple shapes like rectangles or trapezoids and is used in integrals when finite sums are involved. Figuring out the area of a curve is complex hence this method makes it simple. Usually, we take the help of different integration methods for this purpose. This is one of the major parts of integral calculus.
Riemann Integral
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.
Question
![### 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}{](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F08594285-cc6b-4718-bdd1-7a3127ce35f5%2F1af453f3-4d81-4842-b17e-ed8452d8ad75%2F42nuje.png&w=3840&q=75)
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}{
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 2 steps with 2 images
Knowledge Booster
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, calculus and related others by exploring similar questions and additional content below.Recommended textbooks for you
Calculus: Early Transcendentals
Calculus
ISBN:
9781285741550
Author:
James Stewart
Publisher:
Cengage Learning
Thomas' Calculus (14th Edition)
Calculus
ISBN:
9780134438986
Author:
Joel R. Hass, Christopher E. Heil, Maurice D. Weir
Publisher:
PEARSON
Calculus: Early Transcendentals (3rd Edition)
Calculus
ISBN:
9780134763644
Author:
William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz
Publisher:
PEARSON
Calculus: Early Transcendentals
Calculus
ISBN:
9781285741550
Author:
James Stewart
Publisher:
Cengage Learning
Thomas' Calculus (14th Edition)
Calculus
ISBN:
9780134438986
Author:
Joel R. Hass, Christopher E. Heil, Maurice D. Weir
Publisher:
PEARSON
Calculus: Early Transcendentals (3rd Edition)
Calculus
ISBN:
9780134763644
Author:
William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz
Publisher:
PEARSON
Calculus: Early Transcendentals
Calculus
ISBN:
9781319050740
Author:
Jon Rogawski, Colin Adams, Robert Franzosa
Publisher:
W. H. Freeman
Calculus: Early Transcendental Functions
Calculus
ISBN:
9781337552516
Author:
Ron Larson, Bruce H. Edwards
Publisher:
Cengage Learning