Calculus (3rd Edition)

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

Chapter 5 - The Integral - 5.1 Approximating and Computing Area - Exercises - Page 235: 5

Answer

$ R_{5}=44$, $ L_{5}=46$ Average=$\dfrac{44+46}{2}=45$

Work Step by Step

$\Delta x=\frac{b-a}{N}=\frac{1-0}{5}=\frac{1}{5}$ $ f(x_{j})=f(a+j\Delta x)=f(0+j\times\frac{1}{5})=f(\frac{j}{5})$ \begin{equation} R_{N}=\Delta x\sum_{j=1}^{N}f(x_{j}) \end{equation} $\implies $ \begin{equation} R_{5}=\frac{1}{5}\sum_{j=1}^{5}f(\frac{j}{5})=\frac{1}{5}(f(\frac{1}{5})+f(\frac{2}{5})+f(\frac{3}{5})+f(\frac{4}{5})+f(\frac{5}{5}))=\frac{1}{5}(48+46+44+42+40)=44 \end{equation} \begin{equation} L_{N}=\Delta x\sum_{j=0}^{N-1}f(x_{j}) \end{equation} $\implies $ \begin{equation} L_{5}=\frac{1}{5}\sum_{j=0}^{4}f(x_{j})=\frac{1}{5}(f(\frac{0}{5})+f(\frac{1}{5})+f(\frac{2}{5})+f(\frac{3}{5})+f(\frac{4}{5}))=\frac{1}{5}(50+48+46+44+42)=46 \end{equation} Average=$\dfrac{44+46}{2}=45$
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