Answered: 4. The cylinder with equation x² + y² =…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

4. The cylinder with equation x² + y² = 1 bounded by 0 ≤ z ≤ 1 is parameterized by
r(u, v): (cos u, sin u, v),
u € [0, 2π], v = [0, 1].
Modify this parameterization in the following ways:
(a) Rotate the cylinder to be centered around the y-axis instead: the result should be the
cylinder with equation x² + ² = 1 bounded by 0 ≤ y ≤ 1.
(b) Shift the cylinder by 1 unit in the y-direction: the result should be the cylinder with
equation x² + (y - 1)² = 1 bounded by 0 ≤ z ≤ 1.
(c) Make the cylinder 4 times wider and 2 times longer.

Expert Solution

Step 1: Given

The equation of the given cylinder is $x squared plus y squared equals 1$ bounded by $0 less or equal than z less or equal than 1$ is parameterized by

$r open parentheses u comma v close parentheses equals open parentheses cos u comma space sin u comma space v close parentheses comma space space space u element of open square brackets 0 comma 2 pi close square brackets comma space v element of open square brackets 0 comma 1 close square brackets$

To modify the parametrization as follows

(a) Rotate the cylinder to be centered around the $y$ -axis instead: the result should be the cylinder with equation $x squared plus z squared equals 1$ bounded by $0 less or equal than y less or equal than 1$ .

(b) To shift the cylinder by one unit in the $y$ -direction: the result should be the cylinder with equation $x squared plus open parentheses y minus 1 close parentheses squared equals 1$ bounded by $0 less or equal than z less or equal than 1$ .