Calculus, 10th Edition (Anton)

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

Chapter 4 - Integration - 4.8 Average Value Of A Function And It's Applications - Exercises Set 4.8 - Page 335: 13

Answer

See explanation.

Work Step by Step

Since the graph of $f$ is a linear function, the area between $f$ and the $x$ -axis has a trapezoidal shape and is $\frac{(-a+b) \cdot (f(b)+f(a)) }{2}$ The average value of $f$ on $[a, b]$ is $f_{a v \epsilon}= \frac{(f(a)+f(b)) \cdot(-a+b)}{2} \cdot \frac{1}{-a+b} =$ $ \frac{(f(b)+f(a)) }{2}$ Since $f$ is a linear function, we have $f_{a v g}=\frac{f(b)+f(a)}{2}=f\left(\frac{b+a}{2}\right)$
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