Calculus (3rd Edition)

Published by W. H. Freeman
ISBN 10: 1464125260
ISBN 13: 978-1-46412-526-3

Chapter 6 - Applications of the Integral - 6.4 The Method of Cylindrical Shells - Exercises - Page 312: 34


$12 \pi$

Work Step by Step

The shell method to compute the volume of a region: The volume of a solid obtained by rotating the region under $y=f(x)$ over an interval $[m,n]$ about the y-axis is given by: $V=2 \pi \int_{m}^{n} (Radius) \times (height \ of \ the \ shell) \ dy=2 \pi \int_{m}^{n} (x) \times f(x) \ dx$ Now, $V=2\pi \int_{0}^{1} 3x dx +2 \pi \int_1^4 (x) \sqrt {4x^{-1}-1} \ dx \\ = 2\pi \int_0^1 3x dx + 2 \pi \int_1^4 (4-x) \ dx \\=2\pi [\dfrac{3x^2}{2}]_0^1+2 \pi[4x-\dfrac{x^2}{2}]_1^4 \\ = 2\pi \times \dfrac{3}{2} +2 \pi (8-\dfrac{7}{2}) \\ =12 \pi$
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