Calculus (3rd Edition)

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

Chapter 14 - Calculus of Vector-Valued Functions - 14.1 Vector-Valued Functions - Exercises - Page 710: 16


See the explanation below.

Work Step by Step

The projection onto the xy-plane is traced by $\langle \sin t,0\rangle $ which is a segment $[-1,1] $ on the x-axis since $-1\leq \sin t \leq 1$. The projection onto the xz-plane is the circle itself. We put $$ x=\sin t, z-4=\cos t $$ hence we get $$ x^2+(z-4)^2=1$$ which is a circle of radius $1$ centered at $(0,0,4)$. The projection onto the yz-plane is traced by $\langle 0,0,4+\cos t\rangle $ which is a segment $[3,5] $ on the z-axis since $3=4-1\leq 4+ \cos t \leq 4+1=5$.
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