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: 52

Answer

$$0$$

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}\frac{2}{n} \cdot \left(\frac{2 k}{n}-1\right)^{3} $ $x_{k}^{*}$ is the left endpoint, $f(x)=x^{3}, \Delta x=\frac{2}{n}$ Rewrite sum in closed form: \[ A=\lim _{n \rightarrow+\infty} \frac{2 n^{2}}{n^{3}} \] Evaluate limit \[ A=0 \]
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