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 365: 37


a) $$\dfrac{9 \pi}{16}$$ b) $$\dfrac{9 \pi}{16}$$

Work Step by Step

a) $$Volume= (\pi) \int_{1/16}^{1} (x^{-1/4})^2)-(1)^2 \space dx \\=2 \pi \sqrt {x}-\pi y \\=\dfrac{9 \pi}{16}$$ b) Consider the shell model to compute the volume: $$Volume=\int_{m}^{n} (2 \pi) (\space Radius \space of \space shell) \times ( height \space \text{of} \space \text {shell}) dx \\= (2 \pi) \int_{1}^{2}(y) (\dfrac{1}{y^4}-\dfrac{1}{16}) \\=\dfrac{9 \pi}{16}$$
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