University Calculus: Early Transcendentals (3rd Edition)

Published by Pearson
ISBN 10: 0321999584
ISBN 13: 978-0-32199-958-0

Chapter 5 - Section 5.3 - The Definite Integral - Exercises - Page 310: 77


$\int_a^b f(x) dx \geq 0$

Work Step by Step

When the function $f(x)$ has minimum and maximum on $[a,b]$ then, we have $min f(b-a) \leq \int_a^b f(x) dx \leq max f(b-a)$ ...(1) Here, we have the function $f(x) \geq 0$ on $[a,b]$ then $ min f \geq 0$ and $ max f \geq 0$ on $[a,b]$ Then, we have $(b-a) \geq0; (b-a) min f \geq 0$ and $(b-a) max f \geq 0$ From the equation (1), we get that both sides are positive. Hence, the result has been verified $\int_a^b f(x) dx \geq 0$
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