Calculus: Early Transcendentals (2nd Edition)

Published by Pearson
ISBN 10: 0321947347
ISBN 13: 978-0-32194-734-5

Chapter 5 - Integration - 5.4 Working with Integrals - 5.4 Exercises: 9

Answer

$$\frac{{1000}}{3}$$

Work Step by Step

$$\eqalign{ & \int_{ - 2}^2 {\left( {3{x^8} - 2} \right)dx} \cr & {\text{split the integral}} \cr & \int_{ - 2}^2 {3{x^8}dx} - \int_{ - 2}^2 {2dx} \cr & 3{x^8}{\text{ is an even function}}{\text{, 2 is an even function}} \cr & {\text{Use the theorem 5}}{\text{.4}} \cr & {\text{If }}f\left( x \right){\text{ is even}}{\text{, }}\int_{ - a}^a {f\left( x \right)dx} = 2\int_0^a {f\left( x \right)dx} \cr & = 2\int_0^2 {3{x^8}dx} - 2\int_0^2 {2dx} \cr & {\text{integrating}} \cr & = 2\left. {\left( {\frac{{3{x^9}}}{9}} \right)} \right|_0^2 - 2\left. {\left( {2x} \right)} \right|_0^2 \cr & = 2\left. {\left( {\frac{{{x^9}}}{3} - 2x} \right)} \right|_0^2 \cr & {\text{using the fundamental theorem}} \cr & = 2\left( {\frac{{{{\left( 2 \right)}^9}}}{3} - 2\left( 2 \right)} \right) - 2\left( {\frac{{{{\left( 0 \right)}^9}}}{3} - 2\left( 0 \right)} \right) \cr & = 2\left( {\frac{{500}}{3}} \right) \cr & = \frac{{1000}}{3} \cr} $$
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