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: 6


$$36 \pi$$

Work Step by Step

Consider the shell model to compute the volume: $$V=\int_{m}^{n} (2 \pi) (\space Radius \space of \space shell) \times ( height \space \text{of} \space \text {shell}) dx \\ = \int_0^{3} (2 \pi) (x) \times [\dfrac{9x}{(x^3+9)^{1/2}}) \space dx$$ Let us consider $a=x^3+9$ and $ da=3x^2dx$ $$V=(2 \pi) \times \int_{9}^{36} (3) \times a^{-(1/2)} da \\=6 \pi \times (2 \sqrt a)]_{9}^{36} \\=36 \pi$$
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