Calculus: Early Transcendentals 9th Edition

Published by Cengage Learning
ISBN 10: 1337613924
ISBN 13: 978-1-33761-392-7

Chapter 5 - Review - Exercises - Page 430: 49

Answer

a) $2$ b) $6$

Work Step by Step

Recall: If $f(x)$ is above (or below) the axis for $a\leq x\leq b$ and $A$ is the area bounded by the curve of $f$ and the $x-$axis for $a\leq x\leq b$, then $A=\int_a^bf(x)dx$ (or $A=-\int_a^bf(x)dx$). Using this knowledge above, we have $\int_1^3f(x)dx=A$, $\int_3^4 f(x)dx=-B$, and $\int_4^5f(x)dx=C$. Part a) Using the properties for Integrals, $\int_1^5 f(x) dx=\int_1^3f(x)dx+\int_3^4 f(x)dx+\int_4^5f(x)dx=A+(-B)+C=3+(-2)+1=2$ Part b) Using the properties for Integrals, $\int_1^5 |f(x)| dx=\int_1^3|f(x)|dx+\int_3^4 |f(x)|dx+\int_4^5|f(x)|dx$ $=\int_1^3f(x)dx+\int_3^4-f(x)dx+\int_4^5f(x)dx$ $=\int_1^3f(x)dx+(-\int_3^4 f(x)dx)+\int_4^5f(x)dx$ $=A+B+C$ $=3+2+1$ $=6$
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