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: 36

Answer

$$\frac{25}{2}$$

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{5}{n} \cdot \left(-\frac{5 k}{n}+5\right) $ $x_{k}^{*}$ is the right endpoint, $f(x)=-x+5, \Delta x=\frac{5}{n}$ $A=\lim _{n \rightarrow+\infty} \frac{25}{n} \sum_{k=1}^{n} -\frac{25}{n^{2}+1} \sum_{k=1}^{n} k$ $=\lim _{n \rightarrow+\infty}\left(\frac{25 n}{n}-\frac{(1+n)25 n}{2 n^{2}}\right)$ $=\lim _{n \rightarrow+\infty}\left(\frac{25 n+25 n^{2}}{2 n^{2}}\right)$ Rewrite sum in closed form \[ \frac{25}{2}=A \]
Update this answer!

You can help us out by revising, improving and updating this answer.

Update this answer

After you claim an answer you’ll have 24 hours to send in a draft. An editor will review the submission and either publish your submission or provide feedback.