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 363: 11



Work Step by Step

Consider the shell model to compute the volume: $$Volume=\int_{m}^{n} (2 \pi) (\space Radius \space of \space shell) \times ( height \space \text{of} \space \text {shell}) dx \\= \int_0^{1} (2 \pi) \cdot (x)[\sqrt x-(2x-1)] dx \\ =2 \pi [(\dfrac{2}{5})x^{5/2}-(\dfrac{2}{3})x^3+\dfrac{x^2}{2}]_0^1\\ = 2 \pi \times (\dfrac{2}{5}-\dfrac{2}{3}+\dfrac{1}{2}) \\=\dfrac{7\pi}{15}$$
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