James Stewart-Calculus_ Early Transcendentals-Cengage Learning (2015)

section 16.6 Parametric Surfaces and Their Areas 1121
;
;
;
;
26. The part of the plane z − x 1 3 that lies inside the
cylinder x 2 1 y 2 − 1
27–28 Use a graphing device to produce a graph that looks like
the given one.
27. 28.
3
z 0
_3
0
_3
0 5 x
y
0
z
_1
_1
0
y
1
1
0
x
29. Find parametric equations for the surface obtained by
rotating the curve y − 1ys1 1 x 2 d, 22 < x < 2, about
the x-axis and use them to graph the surface.
30. Find parametric equations for the surface obtained by
rotating the curve x − 1yy, y > 1, about the y-axis and
use them to graph the surface.
31. (a) What happens to the spiral tube in Example 2 (see Figure
5) if we replace cos u by sin u and sin u by cos u?
(b) What happens if we replace cos u by cos 2u and sin u
by sin 2u?
; 32. The surface with parametric equations
CAS
x − 2 cos 1 r cossy2d
y − 2 sin 1 r cossy2d
z − r sinsy2d
where 2 1 2 < r < 1 2 and 0 < < 2, is called a Möbius
strip. Graph this surface with several viewpoints. What is
unusual about it?
33–36 Find an equation of the tangent plane to the given parametric
surface at the specified point.
33. x − u 1 v, y − 3u 2 , z − u 2 v; s2, 3, 0d
34. x − u 2 1 1, y − v 3 1 1, z − u 1 v; s5, 2, 3d
35. rsu, vd − u cos v i 1 u sin v j 1 v k; u − 1, v − y3
36. rsu, vd − sin u i 1 cos u sin v j 1 sin v k;
u − y6, v − y6
37–38 Find an equation of the tangent plane to the given
parametric surface at the specified point. Graph the surface and
the tangent plane.
37. rsu, vd − u 2 i 1 2u sin v j 1 u cos v k; u − 1, v − 0
38. rsu, vd − s1 2 u 2 2 v 2 d i 2 v j 2 u k; s21, 21, 21d
_1
CAS
CAS
39–50 Find the area of the surface.
39. The part of the plane 3x 1 2y 1 z − 6 that lies in the
first octant
40. The part of the plane with vector equation
rsu, vd − ku 1 v, 2 2 3u, 1 1 u 2 vl that is given by
0 < u < 2, 21 < v < 1
41. The part of the plane x 1 2y 1 3z − 1 that lies inside the
cylinder x 2 1 y 2 − 3
42. The part of the cone z − sx 2 1 y 2 that lies between the
plane y − x and the cylinder y − x 2
43. The surface z − 2 3 sx 3y2 1 y 3y2 d, 0 < x < 1, 0 < y < 1
44. The part of the surface z − 4 2 2x 2 1 y that lies above the
triangle with vertices s0, 0d, s1, 0d, and s1, 1d
45. The part of the surface z − xy that lies within the
cylinder x 2 1 y 2 − 1
46. The part of the surface x − z 2 1 y that lies between the
planes y − 0, y − 2, z − 0, and z − 2
47. The part of the paraboloid y − x 2 1 z 2 that lies within the
cylinder x 2 1 z 2 − 16
48. The helicoid (or spiral ramp) with vector equation
rsu, vd − u cos v i 1 u sin v j 1 v k, 0 < u < 1,
0 < v <
49. The surface with parametric equations x − u 2 , y − uv,
z − 1 2 v 2 , 0 < u < 1, 0 < v < 2
50. The part of the sphere x 2 1 y 2 1 z 2 − b 2 that lies inside the
cylinder x 2 1 y 2 − a 2 , where 0 , a , b
51. If the equation of a surface S is z − f sx, yd, where
x 2 1 y 2 < R 2 , and you know that | fx | < 1 and | fy | < 1,
what can you say about AsSd?
52–53 Find the area of the surface correct to four decimal
places by expressing the area in terms of a single integral and
using your calculator to estimate the integral.
52. The part of the surface z − cossx 2 1 y 2 d that lies inside the
cylinder x 2 1 y 2 − 1
53. The part of the surface z − lnsx 2 1 y 2 1 2d that lies above
the disk x 2 1 y 2 < 1
54. Find, to four decimal places, the area of the part of the
surface z − s1 1 x 2 dys1 1 y 2 d that lies above the square
| x | 1 | y | < 1. Illustrate by graphing this part of the
surface.
55. (a) Use the Midpoint Rule for double integrals (see Section
15.1) with six squares to estimate the area of the
surface z − 1ys1 1 x 2 1 y 2 d, 0 < x < 6, 0 < y < 4.
(b) Use a computer algebra system to approximate the
surface area in part (a) to four decimal places. Compare
with the answer to part (a).
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