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: 21


$$\dfrac{ 16\pi}{3}$$

Work Step by Step

Consider the shell model to compute the volume: $$Volume=\int_{m}^{n} (2 \pi) (\space Radius \space of \space shell) \times ( height \space \text{of} \space \text {shell}) dx \\ = \int_{0}^{2} (2 \pi) \cdot y (2+y-y^2) dy \\= (2\pi) [y^2+\dfrac{y^3}{3}-\dfrac{y^4}{4}]_{0}^{2} \\=[4(2 \pi)+\dfrac{8(2 \pi)}{3}-4(2 \pi)] \\=\dfrac{ 16\pi}{3}$$
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