UNIVERSITI TEKNOLOGI MALAYSIA SSE 1893 ENGINEERING MATHEMATICS TUTORIAL 5

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1 UNIVERSITI TEKNOLOGI MALAYSIA SSE 189 ENGINEERING MATHEMATIS TUTORIAL 5 1. Evaluate the following surface integrals (i) (x + y) ds, : part of the surface 2x+y+z = 6 in the first octant. (ii) (iii) (iv) x2 + y 2 + z 2 ds, : the part of the surface of the cone z = x 2 + y 2 from z = 0 to z = 1. ds, : the surface of the cuboid in the first octant bounded by x = 1, y = 1 and z = 1. (x 2 + y 2 ) ds, : the surface of the sphere x 2 + y 2 + z 2 = Evaluate for the given F dan. (i) F(x, y, z) = i + j + k : the part of the surface of the cone z = x 2 + y 2 from z = 0 to z = 1. (ii) F(x, y, z) = (x 2 + y) i + xy j (2xz + y) k : the surface of the tetrahedron in the first octant bounded by x + y + z = 1 and the coordinate axes. (iii) F(x, y, z) = yz j + z 2 k : the surface of y 2 + z 2 = 1, z 0 from x = 0 to x = 1. (iv) F(x, y, z) = x i + y j + z k : the surface of the sphere x 2 + y 2 + z 2 = 4.. Use Gauss s theorem to evaluate for the given F and. (i) F(x, y, z) = (x z) i + (y x) j + (z y) k : the surface of the cylinder x 2 + y 2 = 4, from z = 0 to z = 1. (ii) F(x, y, z) = x i + x 2 y j + xy k : the surface of the region bounded by the parabolic cylinder z = 4 x 2, and the planes y + z = 5 z = 0, y = 0. (iii) F(x, y, z) = x 2 i + y 2 j + z 2 k : the surface of the region bounded by the cone z = x 2 + y 2 dan plane z = 1. (iv) F(x, y, z) = x i + y j + z k : the surface of the cylinder x 2 + y 2 = 9 from z = 1 to z = 4.

2 2 SSM208/228: Tutorial 5 4. Use Stokes s theorem to evaluate ( F). ds for the given F dan. (i) F(x, y, z) = 2x i + xz j + yz k : the part of the surface z = y, between 0 x 2 dan 0 y 2. (ii) F(x, y, z) = xy i + x j + xz k : the part of the surface of the paraboloid z = x 2 + y 2, from z = 0 to the plane z = 4. (iii) F(x, y, z) = z i + 2y j + xz k : the part of the cylindrical surface x 2 + y 2 = 9, y 0 from z = 0 to z = 4. (iv) F(x, y, z) = y i 2x j + xyz k : the part of the surface of the hemisphere z = 4 x 2 y Use Stokes s theorem to evaluate for the given F dan. (i) F(x, y, z) = x 2 i + 4xy j + xy 2 k : the closed curve on the plane z = y with vertices (1, 0, 0), (1,, ), (0,, ) dan (0, 0, 0) with the orientation counterclockwise looking from above. (ii) F(x, y, z) = (x 2 y) i + 4z j + x 2 k : the circle of intersection between the plane z = 2 and the surface of the cone z = x 2 + y 2 with the orientation counterclockwise looking from above. (iii) F(x, y, z) = xz i + xy j + xz k : the line segment around the plane 2x+y +z = 2 that lies in the first octant with the orientation counterclockwise looking from above. (iv) F(x, y, z) = y i 2x j 4x k : the ellipse obtained from the intersection between the cylinder x 2 + y 2 = 4 and the plane z = x with the orientation counterclockwise looking from above. 6. Session 2001/02 Sem I a) Given F(x, y, z) = x i y j + x 2 k and is the surface of the solid bounded by the cone z = x 2 + y 2 and the plane z = 1 and z = 2. By using the Gauss s theorem, obtain if F(x, y, z) = ( y 2 ) i (4xy) j + (yz) k, and is the curve around the plane x + y + z = 1 that lies in the first octant with the orientation counterclockwise looking from above.

3 SSM208/228: Tutorial 5 7. Session 2001/02 Sem II a) Evaluate with F(x, y, z) = (2x y) i (yz 2 ) j (y 2 z) k, and is the surface of a paraboloid z = 4 x 2 y 2 located above the xy plane. heck your answer by using the Stokes s theorem. b) Let F(x, y, z) = x 2 i + y 2 j + z 2 k and is the boundary of the solid inside a cylinder x 2 + y 2 = 4 and the plane z = 0 and z = 2. Use Gauss s theorem to evaluate with n as the unit normal vector to pointing outwards. 8. Session 2002/0 Sem I a) By using the Gauss s theorem, evaluate with F(x, y, z) = (x + tan yz) i + (y e xz ) j + (z + x ) k and is the surface of a hemisphere z = 4 x 2 y 2. if F(x, y, z) = (2z) i + (8x y) j + (x + y) k, and is the closed curve around the triangular plane that lies in the first octant with vertices (1, 0, 0), (0, 1, 0), and (0, 0, 2) in the orientation counterclockwise looking from above. 9. Session 2002/0 Sem II a) Let G be the solid bounded by the surfaces x 2 + y 2 = 9, z = 0, and z =. By using the Gauss s theorem, evaluate with F(x, y, z) = x i + y j + z k and is the surface of G with the unit normal vector n pointing outwards.

4 4 SSM208/228: Tutorial 5 if F(x, y, z) = (y 2 +z 2 ) i + (x 2 +z 2 ) j + (x 2 ) k, and is the closed curve around the plane x+y+z = 1 that lies in the first octant in the orientation counterclockwise looking from above. 10. Session 2005/06 Sem I a) Evaluate z ds if is the portion of the cone z = 2 x 2 + y 2 between the planes z = 2 and z = 4. b) Use Gauss s theorem, evaluate if F(x, y, z) = x i +2y j +z k and is the surface of a tetrahedron bounded by the plane x+y+z = 1 in the first octant. c) Given F(x, y, z) = (x 2 + y 2 ) i + (4z) j + (x 2 ) k, and is the circular curve of the intersection between the plane z = 0 and the surface of the paraboloid z = 4 x 2 y 2 in the orientation counterclockwise looking from above. Use Stokes s theorem to evaluate 11. Session 2006/07 Sem I (6 Marks) a) Find the surface area of the paraboloid z = 2(x 2 + y 2 ) below the plane z = 8. b) Let be the surface of a solid G oriented by outward unit normals, and V is the volume of G. If F(x, y, z) = x i + y j + z k and n is an outer unit normal to, use Gauss s theorem to show that V = 1 Using this formula, find the volume of the cone z = x 2 + y 2 bounded above by the plane z = 2. c) Use Stokes s theorem to evaluate, where F(x, y, z) = y 2 i + z 2 j + x 2 k, and is the triangle with vertices P (1, 0, 0), Q(0, 1, 0), and R(0, 0, 1) traversed counterclockwise as seen from above. (6 Marks)

5 SSM208/228: Tutorial Session 2007/08 Sem II a) Evaluate yz 2 ds where is the portion of the curved surface of the cylinder x 2 + z 2 = 4 in the first octant between y = 0 and y = 5., (5 Marks) where F(x, y, z) = (z sin x) i + (x 2 + e y ) j + (y cos z) k, and is the closed curve enclosing the surface of the disk x 2 + y 2 = 9 on the xy plane. c) Suppose F(x, y, z) = x i + y j + z k and is the closed surface of the region bounded by the hemisphere z = 4 x 2 y 2 and the plane z = 0. Use Gauss s theorem, evaluate ANSWERS FOR TUTORIAL 5 1. (i) 5 14 (ii) 4π (iii) 6 (iv) 128π 1 2. (i) π (ii) (iii) 2 (iv) 2π 24. (i) 12π (ii) 11 2 (iii) π (iv) π 4. (i) 8 (ii) 4π (iii) 24 (iv) 20π 5. (i) 90 (ii) 4π (iii) 1 (iv) 20π (8 Marks) 6. (a) 8π (b) (a) 4π (b) 16π 8. (a) 41 5 π (b) 4 9. (a) 180π (b) (a) 28 5 π (b) 1 (c) (a) (65 2 1)π 24 (b) 1 πa (c) (a) 25π (b) 0 (c) π