Calculus 8th Edition

Published by Cengage
ISBN 10: 1285740629
ISBN 13: 978-1-28574-062-1

Chapter 16 - Vector Calculus - 16.6 Parametric Surfaces and Their Areas - 16.6 Exercises - Page 1160: 23


$x=2 \sin \phi, \cos \theta; y=2 \sin \phi, \sin \theta;z=2 \cos \phi$ ;$0 \leq \phi \leq \pi/4,0 \leq \theta \leq \pi/4$ or, $x=x,y=y,z=\sqrt{4-x^2-y^2}; x^2+y^2 \leq \sqrt 2$

Work Step by Step

Consider the parametric equations inside the cone are: $( r \cos \theta, r \sin \theta , z)$ The equation of sphere as $z=\sqrt{4-(x^2+y^2)}=\sqrt{4-(r^2 \sin^2 \theta+r^2 \cos \theta)}=\sqrt {4-r^2}$ Now, the points inside the sphere becomes: $[ r \cos \theta, r \sin \theta , \sqrt{4-(x^2+y^2)}] ; \theta \in [0, 2 \pi] $ and $r \in [\sqrt 2,2] $ Find the range of $r$. $\sqrt{x^2+y^2}=\sqrt{4-(x^2+y^2)}$ $\implies x^2+y^2=4-(x^2+y^2)$ That is, $r=\sqrt 2$ So, the parametric equations or representation for the sphere: $x=2 \sin \phi, \cos \theta; y=2 \sin \phi, \sin \theta;z=2 \cos \phi$ ;$0 \leq \phi \leq \pi/4,0 \leq \theta \leq \pi/4$ or, $x=x,y=y,z=\sqrt{4-x^2-y^2}; x^2+y^2 \leq \sqrt 2$
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