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: 40

Answer

$$ 22 $$

Work Step by Step

Definition 5.4.3 $A=\lim _{n \rightarrow+\infty} \sum_{k=1}^{n} f\left(x_{k}^{*}\right) \Delta x$ $x_{k}^{*}$ is the left endpoint, so: $A=\lim _{n \rightarrow+\infty} \sum_{k=1}^{n}\left(1-\left(-3+\frac{2 k}{n}\right)^{3}\right) \cdot \frac{2}{n}$ $f(x)=-x^{3}+1, \Delta x=\frac{2}{n}$ Rewrite sum in closed form: \[ A=\lim _{n \rightarrow+\infty} \frac{22 n^{2}-26 n+8}{n^{2}} \] $A=22$
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