1 2 z 21 16x 2 4x 2 y2 y2 4 16z 2 4z 2 1 4 3 2 z

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Unformatted text preview: z 4 z2 z 4 2 x2 3 5 π x 3 y −2 −1 1 2 x y 4 x 4 y 37. z 4 z 5 xy 39. 4x 2 z ± y2 4z 2 y2 4 z 8 16 x2 4 41. z z 2 x2 2 y2 y2 z y2 2 x2 x2 −4 −6 −8 3 2 1 5 x 4 3 3 4 5 y −2 8 x 6 4 6 4 −2 −4 −6 −8 2 2 y −2 −2 2 x 1 2 y 43. x2 x z y2 z 0 2 1 4 3 2 z 45. x2 z2 x2 z2 ry 2 and z ry ±2 y ; therefore, 4y. 3 x 2 3 y 47. x2 y2 x2 y2 rz 2 and y rz 4y 2 z ; therefore, 2 z 2. 49. y 2 z2 y2 z2 rx 2 and y rx z2 2 ; therefore, x 4 . x2 z2 2 , 4x 4 0 2z 2 22 2 ,y x 51. x 2 x2 y2 y2 2z 53. Let C be a curve in a plane and let L be a line not in a parallel plane. The set of all lines parallel to L and intersecting C is called a cylinder. 2z or x 2z z Equation of generating curve: y 4 55. See pages 765 and 766. 57. V 2 0 x 4x 4x3 3 x4 4 x2 dx 4 0 4 3 2 1 2 128 3 h ( x) x 1 2 3 4 p ( x) 252 Chapter 10 x2 2 y2 4 Vectors and the Geometry of Space 59. z (a) When z 2 we have 2 42 4 4, c x2 2 y2 , or 1 4 x2 4 y2 8 (b) When z 8 we have 8 x2 2 y2 , or 1 4 x2 16 y2 . 32 Major axis: 2 8 Minor axis: 2 4 c2 a2 b2, c2 Foci: 0, ± 2, 2 Major axis: 2 32 Minor axis: 2 16 2 c2 32 16 Foci: 0, ± 4, 8 82 8 4 16, c 61. If x, y, z is on the surface, then y y2 4y x2 2 2 63. z2 4 z 2 x2 39632 y2 39632 z z2 39422 1 x2 x 2 y y 2 2)2 4y 4 z2 4000 8y 4000 Elliptic paraboloid Traces parallel to xz-plane are circles. 4000 x y 65. z y2 b2 x2 ,z a2 bx bx ay a4b2 4 a2b 2 a2 y 2 ay y2 b2 12 y b2 y x2 a2 ab2y ab2 2 b2 ± 2 67. The Klein bottle does not have both an “inside” and an “outside.” It is formed by inserting the small open end through the side of the bottle and making it contiguous with the top of the bottle. a2b4 4 12 x a2 a2 bx x b x a a2b 2 ab2 2 Letting x x at, y z 2abt at, you obtain the two intersecting lines bt, z 0 and x at, y bt ab2 a2b2. Section 10.7 1. 5, 0, 2 , cylindrical x y z 5 cos 0 5 sin 0 2 5 0 Cylindrical and Spherical Coordinates 3. x y z 1, 2, , 2 , cylindrical 3 2 cos 2 sin 2 3, 2 , rectangular 3 3 1 3 5. x y z 4, 7 , 3 , cylindrical 6 4 cos 4 sin 3 2 3, 2, 3 , rectan...
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