Calculus, 10th Edition (Anton)

Published by Wiley
ISBN 10: 0-47064-772-8
ISBN 13: 978-0-47064-772-1

Chapter 4 - Integration - 4.4 The Definition Of Area As A Limit; Sigma Notation - Exercises Set 4.4 - Page 298: 54

Answer

$$\frac{2}{3}$$

Work Step by Step

Definition 5.4.3 $A=\lim _{n \rightarrow+\infty} \sum_{k=1}^{n} f\left(x_{k}^{*}\right) \Delta x$ $A=\lim _{n \rightarrow+\infty} \sum_{k=1}^{n}\left(\frac{k}{n}\right)^{2} \cdot \frac{1}{n}$ $x_{k}^{*}$ is the left endpoint, $f(x)=x^{2}, \Delta x=\frac{1}{n}$ Rewrite sum in closed form: \[ A=\lim _{n \rightarrow+\infty} \sum_{k=1}^{n} \frac{(1+2 n)(1+n)}{6 n^{2}} \] \[ A=\frac{1}{3} \] Evaluate limit \[ B=-\frac{1}{3}+1=\frac{2}{3} \]
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