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: 47


$\dfrac{625 \pi }{6} $

Work Step by Step

We calculate the volume using the disk method: $V=\pi \int_{m}^{n} (R^2) \ dy \\=\pi \int_0^5 [y(5-y)] \ dy \\=\pi \int_0^5 y^2(5-y)^2 \ dy \\=\pi \int_0^5 y^2 (25-10y+y^2) \ dy \\= \pi \int_0^5 (25y^2-10y^3+y^4) \ dy _0^2 \\=\pi [\dfrac{25y^3}{3}-\dfrac{10y^4}{4}+\dfrac{y^5}{5}]_0^5 \\=\dfrac{625 \pi }{6} $
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