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: 9


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

Work Step by Step

Consider the shell model to compute the volume: $$V=\int_{m}^{n} (2 \pi) (\space Radius \space of \space shell) \times ( height \space \text{of} \space \text {shell}) dx \\= \int_0^{2} (2 \pi) \cdot (x)[2-x-x^2] \space dx \\ =2 \pi \times [x^2-\dfrac{x^3}{3}-\dfrac{x^4}{4}]_0^1 \\= (2 \pi) (1-\dfrac{1}{3}-\dfrac{1}{4}) \\=\dfrac{5 \pi}{6}$$
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