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: 70



Work Step by Step

Apply formula:$\int_{A}^{B} f(x)dx=\lim\limits_{n \to \infty} \Sigma_{k=1}^nf(a+k \triangle x)$ $\int_{0}^{1} 3x-x^3 dx=\lim\limits_{n \to \infty}(\dfrac{1}{n}) \Sigma_{k=1}^n (3)(\dfrac{k}{n})-(\dfrac{k}{n})^3$ Thus, we have $\lim\limits_{n \to \infty}(\dfrac{3}{n^2})\dfrac{n(n+1)}{2}-\dfrac{1}{n^4}\dfrac{n^2(n+1)^2}{4}=\lim\limits_{n \to \infty}(\dfrac{3}{2}) (1+\dfrac{1}{n})-\dfrac{1}{4}(1+\dfrac{1}{n})^2=\dfrac{6-1}{4}=\dfrac{5}{4}$
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