AWL/Thomas_ch16p1143-1228
8/27/04
7:26 4100 AM
Page 1199
16.6 Parametrized Surfaces
1199
EXERCISES 16.6
Finding Parametrizations for Surfaces
In Exercises 116, find a parametrization of the surface. (There are many correct ways to do these, so your answers may not be the same as those in the back of the book.) 4. Cone frustum The portion of the cone z = 2 2x 2 + y 2 between the planes z = 2 and z = 4 2. The paraboloid z = 9 - x 2 - y 2, z 0 6. Spherical cap The portion of the sphere x 2 + y 2 + z 2 = 4 in the first octant between the xy-plane and the cone z = 2x 2 + y 2 7. Spherical band between the planes z = 23> 2 and z = - 23> 2 The portion of the sphere x 2 + y 2 + z 2 = 3 3. Cone frustum The first-octant portion of the cone z = 2x 2 + y 2> 2 between the planes z = 0 and z = 3 1. The paraboloid z = x + y , z 4
2 2
many correct ways to set up the integrals, so your integrals may not be the same as those in the back of the book. They should have the same values, however.) 17. Titled plane inside cylinder The portion of the plane y + 2z = 2 inside the cylinder x 2 + y 2 = 1 18. Plane inside cylinder The portion of the plane z = - x inside the cylinder x 2 + y 2 = 4 19. Cone frustum The portion of the cone z = 2 2x 2 + y 2 between the planes z = 2 and z = 6
5. Spherical cap The cap cut from the sphere x 2 + y 2 + z 2 = 9 by the cone z = 2x 2 + y 2
20. Cone frustum The portion of the cone z = 2x 2 + y 2> 3 between the planes z = 1 and z = 4> 3 21. Circular cylinder band The portion of the cylinder x 2 + y 2 = 1 between the planes z = 1 and z = 4 by the cone z = 2x 2 + y 2
22. Circular cylinder band The portion of the cylinder x 2 + z 2 = 10 between the planes y = - 1 and y = 1 23. Parabolic cap The cap cut from the paraboloid z = 2 - x 2 - y 2 25. Sawed-off sphere The lower portion cut from the sphere x 2 + y 2 + z 2 = 2 by the cone z = 2x 2 + y 2 between the planes z = - 1 and z = 23 24. Parabolic band The portion of the paraboloid z = x 2 + y 2 between the planes z = 1 and z = 4
8. Spherical cap The upper portion cut from the sphere x 2 + y 2 + z 2 = 8 by the plane z = - 2 9. Parabolic cylinder between planes The surface cut from the parabolic cylinder z = 4 - y 2 by the planes x = 0, x = 2 , and z=0 10. Parabolic cylinder between planes The surface cut from the parabolic cylinder y = x 2 by the planes z = 0, z = 3 and y = 2 11. Circular cylinder band The portion of the cylinder y 2 + z 2 = 9 between the planes x = 0 and x = 3 12. Circular cylinder band The portion of the cylinder x 2 + z 2 = 4 above the xy-plane between the planes y = - 2 and y = 2 13. Tilted plane inside cylinder The portion of the plane x + y + z=1 a. Inside the cylinder x 2 + y 2 = 9 b. Inside the cylinder y 2 + z 2 = 9 14. Tilted plane inside cylinder x - y + 2z = 2 a. Inside the cylinder x 2 + z 2 = 3 b. Inside the cylinder y 2 + z 2 = 2 15. Circular cylinder band The portion of the cylinder s x - 2 d2 + z 2 = 4 between the planes y = 0 and y = 3 16. Circular cylinder band The portion of the cylinder y 2 + s z - 5 d2 = 25 between the planes x = 0 and x = 10 The portion of the plane
26. Spherical band The portion of the sphere x 2 + y 2 + z 2 = 4
Integrals Over Parametrized Surfaces
In Exercises 2734, integrate the given function over the given surface. 27. Parabolic cylinder Gs x, y, z d = x, over the parabolic cylinder y = x 2, 0 x 2, 0 z 3 28. Circular cylinder Gs x, y, z d = z, over the cylindrical surface y 2 + z 2 = 4, z 0, 1 x 4 29. Sphere Gs x, y, z d = x 2, over the unit sphere x 2 + y 2 + z 2 = 1 30. Hemisphere Gs x, y, z d = z 2, over the hemisphere x 2 + y 2 + z 2 = a 2, z 0
31. Portion of plane Fs x, y, z d = z, over the portion of the plane x + y + z = 4 that lies above the square 0 x 1, 0 y 1, in the xy-plane 33. Parabolic dome Hs x, y, z d = x 2 25 - 4z, over the parabolic dome z = 1 - x 2 - y 2, z 0 32. Cone Fs x, y, z d = z - x, over the cone 0z1
z = 2x 2 + y 2,
Areas of Parametrized Surfaces
In Exercises 1726, use a parametrization to express the area of the surface as a double integral. Then evaluate the integral. (There are
34. Spherical cap Hs x, y, z d = yz, over the part of the sphere x 2 + y 2 + z 2 = 4 that lies above the cone z = 2x 2 + y 2
Copyright 2005 Pearson Education, Inc., publishing as Pearson Addison-Wesley
4100 AWL/Thomas_ch16p1143-1228
8/27/04
7:27 AM
Page 1200
1200
Chapter 16: Integration in Vector Fields
Flux Across Parametrized Surfaces
In Exercises 3544, use a parametrization to find the flux 4S F # n ds across the surface in the given direction. 35. Parabolic cylinder F = z 2i + xj - 3zk outward (normal away from the x-axis) through the surface cut from the parabolic cylinder z = 4 - y 2 by the planes x = 0, x = 1 , and z = 0 36. Parabolic cylinder F = x 2j - xzk outward (normal away from the yz-plane) through the surface cut from the parabolic cylinder y = x 2, - 1 x 1 , by the planes z = 0 and z = 2 37. Sphere F = zk across the portion of the sphere x 2 + y 2 + z 2 = a 2 in the first octant in the direction away from the origin 38. Sphere F = xi + yj + zk across the sphere x + y + z = a in the direction away from the origin
2 2 2 2
49. Cone The cone rs r, u d = s r cos u di + r s sin u dj + rk, r 0, 0 u 2p at the point P0 A 22, 22, 2 B corresponding to s r, u d = s 2, p> 4 d 51. Circular cylinder The circular cylinder rs u, z d = s 3 sin 2u di + s 6 sin2 u dj + zk, 0 u p, at the point P0 A 3 23> 2, 9> 2, 0 B corresponding to s u, z d = s p> 3, 0 d (See Example 3.)
50. Hemisphere The hemisphere surface rs f, u d = s 4 sin f cos u di + s 4 sin f sin u dj + s 4 cos f dk, 0 f p> 2, 0 u 2p, at the point P0 A 22, 22, 2 23 B corresponding to s f, u d = s p> 6, p> 4 d
52. Parabolic cylinder The parabolic cylinder surface rs x, y d = xi + yj - x 2k, - q 6 x 6 q , - q 6 y 6 q , at the point P0s 1, 2, - 1 d corresponding to s x, y d = s 1, 2 d
39. Plane F = 2xyi + 2yzj + 2xzk upward across the portion of the plane x + y + z = 2a that lies above the square 0 x a, 0 y a , in the xy-plane 40. Cylinder F = xi + yj + zk outward through the portion of the cylinder x 2 + y 2 = 1 cut by the planes z = 0 and z = a 41. Cone F = xyi - zk outward (normal away from the z-axis) through the cone z = 2x 2 + y 2, 0 z 1
2
Further Examples of Parametrizations
53. a. A torus of revolution (doughnut) is obtained by rotating a circle C in the xz-plane about the z-axis in space. (See the accompanying figure.) If C has radius r 7 0 and center (R, 0, 0), show that a parametrization of the torus is rs u, y d = ss R + r cos u dcos y di + ss R + r cos u dsin y dj + s r sin u dk, where 0 u 2p and 0 y 2p are the angles in the figure. b. Show that the surface area of the torus is A = 4p2Rr.
43. Cone frustum F = - xi - yj + z 2k outward (normal away from the z-axis) through the portion of the cone z = 2x 2 + y 2 between the planes z = 1 and z = 2 44. Paraboloid F = 4xi + 4yj + 2k outward (normal way from the z-axis) through the surface cut from the bottom of the paraboloid z = x 2 + y 2 by the plane z = 1
42. Cone F = y i + xzj - k outward (normal away from the zaxis) through the cone z = 2 2x 2 + y 2, 0 z 2
z
C
Moments and Masses
45. Find the centroid of the portion of the sphere x 2 + y 2 + z 2 = a 2 that lies in the first octant. 46. Find the center of mass and the moment of inertia and radius of gyration about the z-axis of a thin shell of constant density d cut from the cone x 2 + y 2 - z 2 = 0 by the planes z = 1 and z = 2. 47. Find the moment of inertia about the z-axis of a thin spherical shell x 2 + y 2 + z 2 = a 2 of constant density d.
0 R
r
u x
48. Find the moment of inertia about the z-axis of a thin conical shell z = 2x 2 + y 2, 0 z 1, of constant density d.
z
Planes Tangent to Parametrized Surfaces
The tangent plane at a point P0s s u0 , y0 d, gs u0 , y0 d, hs u0 , y0 dd on a parametrized surface rs u, y d = s u, y di + gs u, y dj + hs u, y dk is the plane through P0 normal to the vector rus u0 , y0 d * rys u0 , y0 d, the cross product of the tangent vectors rus u0 , y0 d and rys u0 , y0 d at P0. In Exercises 4952, find an equation for the plane tangent to the surface at P0. Then find a Cartesian equation for the surface and sketch the surface and tangent plane together.
y r(u, y)
x u
y
Copyright 2005 Pearson Education, Inc., publishing as Pearson Addison-Wesley
4100 AWL/Thomas_ch16p1143-1228
8/27/04
7:27 AM
Page 1201
16.6 Parametrized Surfaces 54. Parametrization of a surface of revolution Suppose that the parametrized curve C: ((u), g(u)) is revolved about the x-axis, where gs u d 7 0 for a u b . a. Show that rs u, y d = s u di + s gs u dcos y dj + s gs u dsin y dk is a parametrization of the resulting surface of revolution, where 0 y 2p is the angle from the xy-plane to the point r(u, y) on the surface. (See the accompanying figure.) Notice that (u) measures distance along the axis of revolution and g(u) measures distance from the axis of revolution.
y
1201
b. Find a parametrization for the surface obtained by revolving the curve x = y 2, y 0 , about the x-axis. 55. a. Parametrization of an ellipsoid Recall the parametrization x = a cos u, y = b sin u, 0 u 2p for the ellipse s x2> a2 d + s y2> b2 d = 1 (Section 3.5, Example 13). Using the angles u and f in spherical coordinates, show that rs u, f d = s a cos u cos f di + s b sin u cos f dj + s c sin f dk is a parametrization of the ellipsoid s x 2> a 2 d + s y 2> b 2 d + s z 2> c 2 d = 1.
b. Write an integral for the surface area of the ellipsoid, but do not evaluate the integral. 56. Hyperboloid of one sheet a. Find a parametrization for the hyperboloid of one sheet x 2 + y 2 - z 2 = 1 in terms of the angle u associated with the circle x 2 + y 2 = r 2 and the hyperbolic parameter u associated with the hyperbolic function r 2 - z 2 = 1. (See Section 7.8, Exercise 84.) b. Generalize the result in part (a) to the hyperboloid s x2> a2 d + s y2> b2 d - s z2> c2 d = 1.
( f (u), g(u), 0) r (u, y) C y g(u) z f (u) x
57. (Continuation of Exercise 56.) Find a Cartesian equation for the plane tangent to the hyperboloid x 2 + y 2 - z 2 = 25 at the point s x0 , y0 , 0 d, where x02 + y02 = 25. 58. Hyperboloid of two sheets Find a parametrization of the hyperboloid of two sheets s z2> c2 d - s x2> a2 d - s y2> b2 d = 1.
Copyright 2005 Pearson Education, Inc., publishing as Pearson Addison-Wesley