Thomas' Calculus 13th Edition

Published by Pearson
ISBN 10: 0-32187-896-5
ISBN 13: 978-0-32187-896-0

Chapter 6: Applications of Definite Integrals - Section 6.2 - Volumes Using Cylindrical Shels - Exercises 6.2 - Page 331: 33


a) $\dfrac{4 \pi}{15}$ and b ) $\dfrac{7\pi}{30}$

Work Step by Step

a) $V= 2 \pi \int_{0}^{1} (y) \cdot (y-y^3) dy=\dfrac{-2y^3 \pi(3y^2-5)}{15}=\dfrac{4 \pi}{15}$ b) We need to use the shell model as follows: $V=\int_p^{q} (2 \pi) \cdot (\space radius \space of \space shell) ( height \space of \space Shell) \space dx \\= 2 \pi \int_{0}^{1} (1-y)\cdot (y-y^3) dy=\dfrac{2y^2 \pi(12y^3-15y^2-20y+30)}{30}=\dfrac{7\pi}{30}$
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