Calculus (3rd Edition)

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

Chapter 6 - Applications of the Integral - 6.2 Setting Up Integrals: Volume, Density, Average Value - Exercises - Page 298: 37


$\frac{8 \pi}{3} \ {\mathrm{cm}^{3}/\mathrm{s}}$

Work Step by Step

The flow rate is given by $$ 2 \pi \int_{1}^{3} r(0.5)(r-1)(3-r) d r\\ =\pi \int_{1}^{3}\left(-r^{3}+4 r^{2}-3 r\right) d r\\ =\left.\pi\left(-\frac{1}{4} r^{4}+\frac{4}{3} r^{3}-\frac{3}{2} r^{2}\right)\right|_{1} ^{3}\\ =\frac{8 \pi}{3} \ {\mathrm{cm}^{3}/\mathrm{s}} $$
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