Unformatted text preview: z = 2 . 45. Distributing ρ 2 gives x 2 + z 2 = 4 , which is a cylinder with axis on the zaxis. 48. Factoring gives ( ρ4)( ρ2) = 0 , so either ρ = 4 or ρ = 2 . This is the union of two concentric spheres, one of radius 2 and one of radius 4, both centered at the origin. 49. (a) In cylindrical coordinates, this is z = r 2 . (b) In spherical coordinates, this is ρ cos φ = ρ 2 sin 2 φ (cos 2 θ + sin 2 θ ) = ρ 2 sin 2 φ. This can also be rewritten as ρ sin 2 φ = cos φ. 53. (a) In cylindrical coordinates, this hyperboloid of two sheets has equation r 2 (cos 2 θsin 2 θ )2 z 2 = 4 . (b) In spherical coordinates, this is ρ 2 (sin 2 φ cos 2 θsin 2 φ sin 2 θ2cos 2 φ ) = 4 ....
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 Spring '08
 Starr
 Math, Polar coordinate system, cylindrical coordinates

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