Calculus 8th Edition

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

Chapter 14 - Partial Derivatives - 14.1 Functions of Several Variables - 14.1 Exercises: 21


The domain is $$\mathcal{D}=\{(x,y,z)| |x|\leq2,|y|\le3,|z|\le1\}$$ and it is shown on the figure below

Work Step by Step

Arguments of every square root have to be nonnegative which gives $$4-x^2\geq0,\quad 9-y^2\geq0,\quad 1-z^2\geq0.$$ This can be rewritten as $$x^2\leq4,\quad y^2\leq 9,\quad z^2\leq 1.$$ Taking the square root we get $$|x|\leq 2,\quad |y|\leq3,\quad |z|\leq1.$$ Finally, this can be rewritten as $$-2\leq x\leq 2,\quad-3\leq y\leq 3,\quad -1\leq z\leq1,$$ and geometrically, this is a Cuboid bounded by 6 planes $$x=\pm2,\quad y=\pm3,\quad z=\pm1.$$ So we write for the domain $$\mathcal{D}=\{(x,y,z)| |x|\leq2,|y|\le3,|z|\le1\}$$ and it is shown on the figure below
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