galindez (cmg3849) - Homework 13 (7.5,7.6,8.1,8.2) - tegtmeyer - (54045) This print-out should have 16 questions. Multiple-choice questions may continue on the next column or page - find all choices before answering. 001 2. I = 11 32 3. I = 15 32 4. I = 3 8 5. I = 13 32 2.0 points Evaluate the integral 1 I = 4 Z dS S 003 when S is the surface given parametrically by (u, v) = (2uv, u + v, u v) for u2 + v 2 4. Evaluate the integral Z I = 4xyz dS when S is the part of the plane 2z = 2 x + 2y 11 2. I = 3 4. I = 2.0 points S 1. I = 4 3. I = 1 that lies above the rectangle [0, 1] [0, 1] in the xy-plane. 13 3 10 3 1. I = 9 4 2. I = 5 2 5. I = 3 3. I = 3 002 2.0 points Evaluate the integral Z I = f dS S when 5. I = 11 4 004 f (x, y, z) = z 2 + 3xy and S is the portion of the plane x + 2y + 2z = 0 above the unit disk x2 + y 2 1 in the xy-plane. 1. I = 4. I = 2 7 16 2.0 points Calculate the flux of the vector field F(x, y, z) = 3hx + y, x y, x2 + y 2 2zi through the surface S parametrized by (u, v) = u + 2v, u 2v, u2 + 2v 2 with 0 u, v 1, and oriented by u v . 1. I = 6 2. I = 8 galindez (cmg3849) - Homework 13 (7.5,7.6,8.1,8.2) - tegtmeyer - (54045) 2 when S is the part of the paraboloid 3. I = 9 z = 2 x2 y 2 , 4. I = 10 oriented upwards, lying above the square 5. I = 7 005 2.0 points Calculate the flux of the vector field F(x, y, z) = z(x2 + y 2 ), zy, zx 1 x 1 , 1 y 1 , as shown in through the outwardly-oriented open cylinder having radius 1 and lying between the planes z = 0 and z = 2. 1. I = 1. I = 3 2. I = 1 2. I = 3. I = 5 3. I = 2 4. I = 8 3 2 3 4. I = 2 3 2 5. I = 5. I = 2 006 2.0 points Evaluate the integral Z Z I = F dS S for the vector field F = 2x i + y j zk 4 3 007 2.0 points Is the surface integral I = Z Z S F dS positive, negative, or zero for the vector field F = z i and surface S oriented as shown in galindez (cmg3849) - Homework 13 (7.5,7.6,8.1,8.2) - tegtmeyer - (54045) 3 y z x S y x is the curve parametrized by c(t) = (3 cos t 2 sin t, 3 cos t + 2 sin t) , 1. I is zero for 0 t 2. 2. I is negative Use Green's Theorem to determine the area of the shaded region enclosed by the ellipse. 3. I is positive 008 2.0 points Evaluate the integral Z Z I = F dS 1. area = 3 2. area = 6 3. area = 12 S for the vector field 4. area = 12 F = 4x i + 4y j + 4z k , 5. area = 3 when S is the sphere x2 + y 2 + z 2 = 1 , 6. area = 6 oriented outwards. 010 2.0 points 1. I = 4 Use Green's Theorem to evaluate the integral Z I = (2xy 2dx + 3x3 dy) 2. I = 16 3. I = 8 C when C is is the rectangle in the xy-plane having vertices at 4. I = 0 5. I = 12 009 (0, 0), 2.0 points The ellipse shown in 1. I = 21 (2, 0), (2, 1), (0, 1) . galindez (cmg3849) - Homework 13 (7.5,7.6,8.1,8.2) - tegtmeyer - (54045) 4 along the positively oriented boundary of the region between the circles 2. I = 20 x2 + y 2 = 1, 3. I = 23 x2 + y 2 = 16, shown in 4. I = 22 5. I = 24 011 2.0 points Use Green's Theorem to evaluate the integral Z I = (2y dx 2x dy) 1 4 D for the shaded region D shown in y D (not drawn to scale). x 1. I = 2 2. I = 2 3. I = 0 when area(D) = 3. 4. I = 4e8 1. I = 11 5. I = 8 2. I = 12 3. I = 12 013 4. I = 10 When S is the open hemisphere 5. I = 11 x2 + y 2 + z 2 = 1, 6. I = 10 012 2.0 points Use Green's Theorem to evaluate the line integral I = C 3xe2x dx + 2(x4 + 2x2 y 2 ) dy z 0, oriented by the inward normal pointing to the origin, then the boundary orientation on S is counter-clockwise. True or False? 1. TRUE Z 2.0 points 2. FALSE galindez (cmg3849) - Homework 13 (7.5,7.6,8.1,8.2) - tegtmeyer - (54045) 014 5 2.0 points Use Stokes' theorem to evaluate the surface integral Z Z curl F d S S when F(x, y, z) = z 2 sin3 x + y, z 5 + 2x, 3zx4 + y 4 , and S is the northern hemisphere x2 + y 2 + z 2 = 1 , z 0, oriented by the outward normal. formed by the portion of the plane Hint: Find a surface having the same boundary. 1. I = 3 5 2. I = 4 5 3. I = 6 5 in the first octant of 3-space, oriented as shown. 5 6 3 2. I = 2 4 3. I = 3 7 4. I = 6 1. I = 4. I = 5. I = x+y+z = 1 7 5 5. I = 1 015 2.0 points Use Stokes' theorem to evaluate the integral I = Z C F ds 016 2.0 points Use Stokes' theorem to evaluate the integral I = Z Z S when F is the vector field curl F dS when F is the vector field F = 3zx i + 3xy j + yz k and C is the path consisting of the three edges of the triangle ABC shown in F = 2zx i + 3xy j yz k and S is the surface shown in galindez (cmg3849) - Homework 13 (7.5,7.6,8.1,8.2) - tegtmeyer - (54045) whose boundary is the circle c(t) = cos t i + sin t j in the xy-plane. 1. I = 0 2. I = 3 3. I = 1 4. I = 2 5. I = 4 6