Calculus (3rd Edition)

Published by W. H. Freeman
ISBN 10: 1464125260
ISBN 13: 978-1-46412-526-3

Chapter 5 - The Integral - 5.2 The Definite Integral - Exercises - Page 246: 48



Work Step by Step

Since \begin{aligned} R_{N} &=\Delta x \sum_{k=1}^{N} f\left(x_{k}\right)\\ &=\frac{b}{N} \sum_{k=1}^{N}\left(k^{3} \cdot \frac{b^{3}}{N^{3}}\right) \\ &=\frac{b^{4}}{N^{4}} \sum_{k=1}^{N} k^{3}\\ &=\frac{b^{4}}{N^{4}}\left(\frac{N^{4}}{4}+\frac{N^{3}}{2}+\frac{N^{2}}{4}\right) \\ &=\frac{b^{4}}{4}+\frac{b^{4}}{2 N}+\frac{b^{4}}{4 N^{2}} \end{aligned} Then \begin{aligned} \int_{0}^{b} x^{3} d x &=\lim _{N \rightarrow \infty} R_{N}\\ &=\lim _{N \rightarrow \infty}\left(\frac{b^{4}}{4}+\frac{b^{4}}{2 N}+\frac{b^{4}}{4 N^{2}}\right) \\ &=\frac{b^{4}}{4} \end{aligned}
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