Calculus: Early Transcendentals (2nd Edition)

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

Chapter 5 - Integration - 5.3 Fundamental Theorem of Calculus - 5.3 Exercises: 30

Answer

12

Work Step by Step

\[\begin{gathered} \int_0^2 {\left( {3{x^2} + 2x} \right)} dx \hfill \\ \hfill \\ {\text{property }}\int {\left( {f + g} \right)dx} = \int f dx + \int {gdx} \hfill \\ \hfill \\ \int_0^2 {3{x^2}} dx + \int_0^2 {2x} dx \hfill \\ \hfill \\ 3\int_0^2 {{x^2}} dx + 2\int_0^2 x dx \hfill \\ \hfill \\ {\text{Integrate using the rule }}\int {{x^n}} dx = \frac{{{x^{n + 1}}}}{{n + 1}} + C \hfill \\ \hfill \\ 3\left[ {\frac{{{x^3}}}{3}} \right]_0^2 + 2\left[ {\frac{{{x^2}}}{2}} \right]_0^2 \hfill \\ \hfill \\ {\text{Use the fundamental Theorem of calculus}} \hfill \\ \hfill \\ 3\left[ {\frac{{{{\left( 2 \right)}^3}}}{3} - \frac{{{{\left( 0 \right)}^3}}}{3}} \right] + 2\left[ {\frac{{{{\left( 2 \right)}^2}}}{2} - \frac{{{{\left( 0 \right)}^2}}}{3}} \right] \hfill \\ \hfill \\ {\text{Simplify}} \hfill \\ \hfill \\ 3\left[ {\frac{8}{3}} \right] + 2\left[ {\frac{4}{2}} \right] \hfill \\ \hfill \\ 8 + 4 \hfill \\ \hfill \\ 12 \hfill \\ \end{gathered} \]
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