Describe the regions whose equations in cylindrical or spherical coordinates are given below—either in words, or by drawing a picture, as you prefer.(a) The region satisfying the cylindrical conditions r + z ≤ 1 and z ≥ 0.(b) The region satisfying the spherical conditions 0 ≤ ρ ≤ 1, 0 ≤ θ ≤ π, and 0 ≤ ϕ ≤ π.(c) The region satisfying the cylindrical conditions r ≤ 1 and π/4 ≤ θ ≤ 3π/4.(d) The region satisfying the spherical conditions ρ = 1/cos(ϕ) and 0 ≤ ϕ < π/2. Parameterize the following objects:(a) The parallelogram with corners at (0,0,0), (0, 1, 1), (2, 2, 1), and (2,1,0).(Find a paramerization with rectangular domain (u, v) € [a, b] × [c, d].)(b) A "washer" shape in the xy-plane bounded by the inequalities 1< x² + y² ≤ 4.(Find a paramerization with rectangular domain (u, v) € [a, b] × [c, d].)(c) The curve where the cylinder x² + y² = 1 intersects the plane x + y + z = 1.(This is a curve, so find a parameterization with one variable t = [a, b].)

Answered: Image /qna-images/answer/f90e5e50-681a-49d6-82cd-1171275558ba.jpg

Parameterize the following objects: (a) The parallelogram with corners at (0,0,0), (0, 1, 1), (2, 2, 1), and (2,1,0). (Find a paramerization with rectangular domain (u, v) € [a, b] × [c, d].) (b) A "washer" shape in the xy-plane bounded by the inequalities 1< x² + y² ≤ 4. (Find a paramerization with rectangular domain (u, v) € [a, b] × [c, d].) (c) The curve where the cylinder x² + y² = 1 intersects the plane x + y + z = 1. (This is a curve, so find a parameterization with one variable t = [a, b].)

Parameterize the following objects: (a) The parallelogram with corners at (0,0,0), (0, 1, 1), (2, 2, 1), and (2,1,0). (Find a paramerization with rectangular domain (u, v) € [a, b] × [c, d].) (b) A "washer" shape in the xy-plane bounded by the inequalities 1< x² + y² ≤ 4. (Find a paramerization with rectangular domain (u, v) € [a, b] × [c, d].) (c) The curve where the cylinder x² + y² = 1 intersects the plane x + y + z = 1. (This is a curve, so find a parameterization with one variable t = [a, b].)

Elementary Geometry For College Students, 7e

7th Edition

ISBN:9781337614085

Author:Alexander, Daniel C.; Koeberlein, Geralyn M.

Publisher:Alexander, Daniel C.; Koeberlein, Geralyn M.

Chapter10: Analytic Geometry

Section10.6: The Three-dimensional Coordinate System

Problem 41E: Does the sphere x2+y2+z2=100 have symmetry with respect to the a x-axis? b xy-plane?

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