University Calculus: Early Transcendentals (3rd Edition)

Published by Pearson
ISBN 10: 0321999584
ISBN 13: 978-0-32199-958-0

Chapter 5 - Section 5.3 - The Definite Integral - Exercises - Page 310: 64



Work Step by Step

Here, we have $\triangle x=\dfrac{b-a}{n}$ and $c_k=a+\dfrac{k(b-a)}{n}$ Consider the RIEMANN SUM: $\Sigma_{k=1}^n f(c_k) \triangle x=\Sigma_{k=1}^n (\dfrac{8k}{n^2}+\dfrac{2}{n})=\dfrac{8k}{n^2}\Sigma_{k=1}^n k+\dfrac{2}{n}) \Sigma_{k=1}^n$ or, $\dfrac{8k}{n^2}\Sigma_{k=1}^n k+\dfrac{2}{n}) \Sigma_{k=1}^n=\dfrac{4(n+1)}{n}+2$ Now, $\lim\limits_{n \to \infty} \Sigma_{k=1}^n f(c_k) \triangle x=\lim\limits_{n \to \infty}\dfrac{4(n+1)}{n}+2$ Thus, $\lim\limits_{n \to \infty}6+\dfrac{4}{n}=6$
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