Calculus 10th Edition

Published by Brooks Cole
ISBN 10: 1-28505-709-0
ISBN 13: 978-1-28505-709-5

Chapter 7 - Applications of Integration - 7.2 Exercises - Page 453: 17


$\pi(24\ln{4} -\frac{27}{4})$

Work Step by Step

Setup the integration using washer method about the line y=4 $\pi \int_0^3 [(4)^2 - (4-\frac{3}{1+x})^2]dx$ $\pi \int_0^3(16-(16-\frac{24}{1+x}+\frac{9}{(1+x)^2}))dx$ $\pi \int_0^3(\frac{24}{1+x}- \frac{9}{(1+x)^2})dx$ $\pi \int_0^3 (24u^{-1} - 9u^{-2})du$, use u substitution to integrate $\pi [ 24\ln{u} +\frac{9}{u}]_1^4$ , use change of variables and evaluate the definite integral $\pi((24\ln{4} +\frac{9}{4})-(24\ln{1} +9))$ $\pi(24\ln{4} -\frac{27}{4})$
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