Multivariable Calculus, 7th Edition

Published by Brooks Cole
ISBN 10: 0-53849-787-4
ISBN 13: 978-0-53849-787-9

Chapter 15 - Multiple Integrals - 15.3 Exercises - Page 1021: 64

Answer

$\dfrac{2}{3} \pi R^3$

Work Step by Step

Re-write as: $z=\sqrt {R^2-x^2-y^2} $ or, $ z^2=R^2-x^2-y^2$ or, $ x^2+y^2+z^2 =R^2$ The equation of an upper half of a hemi-sphere can be written as: $x^2+y^2+z^2 =R^2$ when $z \geq 0$ So, we can find the volume as follows: $Volume =(\dfrac{1}{2}) (\dfrac{4}{3} \pi R^3) =\dfrac{2 \pi R^3}{3}$
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