Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN: 9781285741550
Author: James Stewart
Publisher: Cengage Learning
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**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.
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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.
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