Homework Help is Here – Start Your Trial Now!
Math
Calculus
Express the limit lim,, +00 Ei-1 8i2 as a definite integral.

Express the limit lim,, +00 Ei-1 8i2 as a definite integral.

Calculus: Early Transcendentals
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...
See similar textbooks
icon
Related questions
Question
**Problem Statement:** Express the limit \(\lim_{n \to \infty} \sum_{i=1}^{n} \frac{8i^2}{n^3}\) as a definite integral. **Explanation:** This mathematical expression involves evaluating the limit of a Riemann sum as \(n\) approaches infinity, which can be expressed as a definite integral. The sum \(\sum_{i=1}^{n} \frac{8i^2}{n^3}\) can be interpreted as approximating the area under a curve by dividing it into \(n\) rectangles, where each rectangle has a width of \(\frac{1}{n}\) and the height given by the function evaluated at that point. To express the sum as a definite integral: 1. Recognize the sum as a Riemann sum for the function \(f(x) = 8x^2\) over the interval from 0 to 1. 2. The definite integral that represents this limit is \(\int_0^1 8x^2 \, dx\). This process of converting sums into integrals is a fundamental concept in calculus, helping to find exact values for areas under curves and other similar problems.
Transcribed Image Text:**Problem Statement:** Express the limit \(\lim_{n \to \infty} \sum_{i=1}^{n} \frac{8i^2}{n^3}\) as a definite integral. **Explanation:** This mathematical expression involves evaluating the limit of a Riemann sum as \(n\) approaches infinity, which can be expressed as a definite integral. The sum \(\sum_{i=1}^{n} \frac{8i^2}{n^3}\) can be interpreted as approximating the area under a curve by dividing it into \(n\) rectangles, where each rectangle has a width of \(\frac{1}{n}\) and the height given by the function evaluated at that point. To express the sum as a definite integral: 1. Recognize the sum as a Riemann sum for the function \(f(x) = 8x^2\) over the interval from 0 to 1. 2. The definite integral that represents this limit is \(\int_0^1 8x^2 \, dx\). This process of converting sums into integrals is a fundamental concept in calculus, helping to find exact values for areas under curves and other similar problems.
Expert Solution
steps

Step by step

Solved in 2 steps with 3 images

SEE SOLUTIONCheck out a sample Q&A here
Blurred answer
Recommended textbooks for you
Calculus: Early Transcendentals
Calculus: Early Transcendentals
Calculus
ISBN:
9781285741550
Author:
James Stewart
Publisher:
Cengage Learning
Thomas' Calculus (14th Edition)
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: Early Transcendentals (3rd Edition)
Calculus
ISBN:
9780134763644
Author:
William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz
Publisher:
PEARSON
Calculus: Early Transcendentals
Calculus: Early Transcendentals
Calculus
ISBN:
9781319050740
Author:
Jon Rogawski, Colin Adams, Robert Franzosa
Publisher:
W. H. Freeman
Precalculus
Precalculus
Calculus
ISBN:
9780135189405
Author:
Michael Sullivan
Publisher:
PEARSON
Calculus: Early Transcendental Functions
Calculus: Early Transcendental Functions
Calculus
ISBN:
9781337552516
Author:
Ron Larson, Bruce H. Edwards
Publisher:
Cengage Learning
SEE MORE TEXTBOOKS