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 365: 44


$$\dfrac{ 4 \pi\space r^3}{3}$$

Work Step by Step

Consider the equation of the circle about the x- axis $r^2=x^2+y^2$ Consider the washer method to compute the volume: $$V=(2) \int_p^{q} (2 \pi) \cdot (\space radius \space of \space shell) ( height \space of \space Shell) \space dy \\= \int_{0}^{r} 2y (\sqrt {r^2-y^2}) \space dy \\=[\dfrac{-4 \pi \times (r^2-y^2)^{(3/2)}}{3}]_{0}^{r} \\=\dfrac{ 4 \space \pi\space r^3}{3}$$
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