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Unformatted text preview: ■ ■ ■ 15y 2
y 2 ■ ■ z 1 for 1 z 2 x 2 5z 2 100 xy
■ ■ ■ sx 2 ■ y2 2. 42. Sketch the region bounded by the paraboloids z x2 y2 2 y. 43. Find an equation for the surface obtained by rotating the parabola y
y x 2 41. Sketch the region bounded by the surfaces z 2z 2
z2 II 0 ■ 5y 2 and z
I ■ 0 4 4z 2y
■ 0 0 2z 2y 2x
■ 20 4z2  Use a computer with threedimensional graphing software to graph the surface. Experiment with viewpoints and with
domains for the variables until you get a good view of the surface. ■ 23. x 2 z
■ 4z 36 3z 2 ; 37–40 21–28 21. x 2 y
■ 4x y2 24z 4y
16 y z2 36. x 4x 2 4z2 ■ x 2y 2 32. 4 x 4 z2 35. x 2 100 4y 2 z 30. x 2 36 3z 2 y2 ■ 17. x 2 9y2 2y 2 33. 4 x 2 y2 16. 25 y 2 4 4x2 31. x z2 x2 14. z 29. z 2 44. Find an equation for the surface obtained by rotating the line y x x 2 about the yaxis. x 3y about the xaxis. 45. Find an equation for the surface consisting of all points that
z III z IV are equidistant from the point
Identify the surface. 1, 0, 0 and the plane x 1. 46. Find an equation for the surface consisting of all points P for
y x which the distance from P to the xaxis is twice the distance
from P to the yzplane. Identify the surface. y
x z V y y x z VII z VI x 47. Show that if the point a, b, c lies on the hyperbolic parabo z VIII loid z y 2 x 2, then the lines with parametric equations
x a t, y b t, z c 2 b a t and x a t,
y b t, z c 2 b a t both lie entirely on this paraboloid. (This shows that the hyperbolic paraboloid is what is
called a ruled surface; that is, it can be generated by the
motion of a straight line. In fact, this exercise shows that
through each point on the hyperbolic paraboloid there are two
generating lines. The only other quadric surfaces that are ruled
surfaces are cylinders, cones, and hyperboloids of one sheet.)
48. Show that the curve of intersection of the surfaces x 2 2y 2 z2
lies in a plane.
x x
■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ 1 and 2 x 2 4y 2 2z2 5y 0 x 2 y 2 and z 1 y 2 on a co...
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 Fall '09
 hamrick

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