Calculus (3rd Edition)

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

Chapter 6 - Applications of the Integral - Chapter Review Exercises - Page 319: 23


$$\frac{3\pi}{2} $$

Work Step by Step

Given $$f(x)=\sqrt{9-x^{2}}, \quad[0,3]$$ The average value is given by \begin{align*} M&=\frac{1}{b-a}\int_{a}^{b}f(x)dx\\ &=\frac{1}{3}\int_{0}^{3}\sqrt{9-x^2}dx \end{align*} Since $$ \int_{0}^{3}\sqrt{9-x^2}dx$$ represents the area of one-quarter of a circle of radius $3$, then$$ \int_{0}^{3}\sqrt{9-x^2}dx=\frac{9\pi}{4} $$ Hence $$M=\frac{9\pi}{2}\frac{1}{3}=\frac{3\pi}{2} $$
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