Calculus, 10th Edition (Anton)

Published by Wiley
ISBN 10: 0-47064-772-8
ISBN 13: 978-0-47064-772-1

Chapter 4 - Integration - 4.6 The Fundamental Theorem Of Calculus - Exercises Set 4.6 - Page 320: 10

Answer

(b) $\sqrt{3}$ (a) $ \pi, \pm 2\pi, \pm 3\pi, \pm 4\pi, ... $

Work Step by Step

Mean-Value Theorem for Integrals: $\int_{a}^{b} f(x) d x=f\left(x^{*}\right)(-a+b)$ (a) $\int_{-\pi}^{\pi} \sin x d x=0$ $\Rightarrow 0= f\left(x^{*}\right).2 \pi$ $\Rightarrow 0=\sin x^{*}=f\left(x^{*}\right)$ $\Rightarrow \pm \pi, \pm 2\pi, \pm 3\pi, \pm 4\pi, ... =x^{*}$ (b) $\int_{1}^{3} \frac{1}{x^{2}} d x=\frac{2}{3}$ $\Rightarrow \frac{2}{3}= f\left(x^{*}\right).2$ $\Rightarrow \frac{1}{3}=\frac{1}{\left(x^{*}\right)^{2}}=f\left(x^{*}\right)$ $\Rightarrow \sqrt{3}=x^{*}$
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