University Calculus: Early Transcendentals (3rd Edition)

Published by Pearson
ISBN 10: 0321999584
ISBN 13: 978-0-32199-958-0

Chapter 6 - Section 6.2 - Volumes Using Cylindrical Shells - Exercises - Page 364: 22


$$\dfrac{ 5\pi}{6}$$

Work Step by Step

Consider the shell model to compute the volume: $$Volume=\int_{m}^{n} (2 \pi) (\space Radius \space of \space shell) \times ( height \space \text{of} \space \text {shell}) dx \\ = (2 \pi) \times \int_{0}^{1} \cdot y (2-y-y^2) dy \\=( 2\pi) [y^2-\dfrac{y^3}{3}-\dfrac{y^4}{4}]_{0}^{1} \\=[-\dfrac{1}{3}(2 \pi)-\dfrac{1}{4}(2 \pi)] \\=\dfrac{ 5\pi}{6}$$
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