Multiple choice questions. Work not needed, just answers. Thanks!
2415 - Quiz 6.tst Write an equivalent double integral with the order of integration reversed. 1) 2 0 4 y2 4y dx dy∫∫ A) 4 0 x 2 4y dy dx∫∫ B) 4 0 x 0 4y dy dx∫∫ C) 2 0 x 0 4y dy dx∫∫ D) 2 0 x 2 4y dy dx∫∫ Express the area of the region bounded by the given line(s) and/or curve(s) as an iterated double integral. 2) The parabola y = x2 and the line y = 3x + 28 A) 7 -4 3x + 28 x2 dy dx∫∫ B) 7 0 3x + 28 x2 dy dx∫∫ C) 7 0 3x + x2 28 dy dx∫∫ D) 7 -4 3x + 28 - x2 0 dx dy∫∫ Evaluate the integral. 3) π/8 0 cos 4x 0 sin 4x dy dx∫∫ A) π 16 B) π 8 C) 1 16 D) 1 8 Integrate the function f over the given region. 4) f(x, y) = xy over the triangular region with vertices (0, 0), (9, 0), and (0, 7) A) 189 8 B) 21 8 C) 147 8 D) 1323 8 Reverse the order of integration and then evaluate the integral. 5) 8 0 1 x/8 ey3 dy dx∫∫ A) 8 3 (e - 1) B) 8 3 (2e - 1) C) 4 3 (2e - 1) D) 4 3 (e - 1) Find the volume of the indicated region. 6) the region bounded by the paraboloid z = 100 - x2 - y2 and the xy-plane A) 10000 3 π B) 5000π C) 5000 3 π D) 2500π Sumit Sarkar Find the center of mass of a thin plate covering the given region with the given density function. 7) The region bounded below by the parabola y = x2 and above by the line y = x + 2, with density ρ(x) = 6x2 A) x = 8 7 , y = 118 49 B) x = 531 70 , y = 18 5 C) x = 18 5 , y = 531 70 D) x = 9 8 , y = 131 64 Evaluate the integral. 8) 2 0 y2 1 z 2 yz dx dz dy∫∫∫ A) - 128 3 B) 23 3 C) 68 3 D) 4 3 Use cylindrical coordinates to find the volume of the indicated region. 9) the region enclosed by the cylinder x2 + y2 = 9 and the planes z = 0 and x + y + z = 6 A) 108π B) 81π C) 27π D) 54π Evaluate the spherical coordinate integral. 10) π/2 0 π/2 0 8 cos φ 0 ρ2 sin φ dρ dφ dθ∫∫∫ A) 64 3 π2 B) 256 9 π C) 64 3 π D) 256 9 π2 Evaluate the line integral of f(x,y) along the curve C. 1) f(x, y) = x2 + y2, C: y = 4x + 2, 0 ≤ x ≤ 3 A) 79 17 B) 543 17 C) 237 D) 237 17 Evaluate the line integral along the curve C. 2) C (xz + y2)∫ ds, C is the curve r(t) = (-7 - 2t)i + tj - 2tk , 0 ≤ t ≤ 1 A) 26 3 B) 26 C) - 16 3 D) - 16 Evaluate the line integral. 3) C y2 dx + x dy∫ ; C is the curve x = t2 - 1, y = 6t, 0 ≤ t ≤ 1 A) 26 B) 14 C) 5 6 D) 72 Find the work done by F over the curve in the direction of increasing t. 4) F = xyi + 8j + 3xk; C: r(t) = cos 8ti + sin 8tj + tk, 0 ≤ t ≤ π 16 A) W = 25 3 B) W = 209 24 C) W = 0 D) W = 193 24 Find the potential function f for the field F. 5) F = (y - z)i + (x + 2y - z)j - (x + y)k A) f(x, y, z) = xy + y2 - xz - yz + C B) f(x, y, z) = x + y2 - xz - yz + C C) f(x, y, z) = x(y + y2) - xz - yz + C D) f(x, y, z) = xy + y2 - x - y + C Evaluate the work done between point 1 and point 2 for the conservative field F. 6) F = 2xi + 2yj + 2zk; P1(2, 2, 5) , P2(3, 6, 9) A) W = 159 B) W = 0 C) W = 93 D) W = - 93 Find the divergence of the field F. 7) F = -6x7i - 8xyj - 4xzk A) -42x6 - 12x B) -54 C) -42x6 - 8y - 4z D) -42x6 - 12x - 54 Find curl F. 8) F(x, y, z) = 2xyzi + 8yzj + 5zk A) -8yi + 2xyj + -2xzk B) -8y + 2xy - 2xz C) 2yzi + 8zj + 5k D) 2yz + 8z + 5 Apply Greenʹs Theorem to evaluate the integral. 9) C (x + 2y)dx + 4xy dy; C the the triangle with vertices (0, 0), (4, 0) and (0, 4). A) 80 B) 160 3 C) 80 3 D) 160 Using Greenʹs Theorem, compute the counterclockwise circulation of F around the closed curve C. 10) F = - 1 8(x2 + y2)4 i; C is the region defined by the polar coordinate inequalities 1 ≤ r ≤ 2 and 0 ≤ θ ≤ π A) 127 448 B) 129 448 C) 0 D) - 127 448 Evaluate the surface integral of f over the surface S. 1) S is the cylinder y2 + z2 = 100, z ≥ 0 and 5 ≤ x ≤ 8; f(x, y, z) = z A) 600 B) 60 C) 300 D) 2600 2) S is the hemisphere x2 + y2 + z2 = 7, z ≥ 0; f(x,y,z) = z2 A) 28 3 π B) 98 3 π C) 196π D) 49 3 π 3) S is the plane x + y + z = 1 above the rectangle 0 ≤ x ≤ 5 and 0 ≤ y ≤ 2; f(x,y,z) = 2z A) 120 B) 80 3 C) - 50 3 D) -80 Find the flux of the vector field F across the surface S in the indicated direction. 4) F = 4xi + 4yj + zk; S is portion of the plane x + y + z = 3 for which 0 ≤ x ≤ 5 and 0 ≤ y ≤ 2; (Oriented upward) A) 270 B) - 75 C) 135 D) 120 Using the Divergence Theorem, find the outward flux of F across the boundary of the region D. 5) F = x2i + y2j + zk; D: the solid cube cut by the coordinate planes and the planes x = 2, y = 2, and z = 2 A) 34 B) 40 C) 24 D) 16 6) F = 2xy2i + 2x2yj + 2xyk ; D: the region cut from the solid cylinder x2 + y2 ≤ 4 by the planes z = 0 and z = 2 A) 128π B) 32π C) 64π D) 16π 7) F = 6x3i + 6y3j + 6z3k ; D: the thick sphere 9 ≤ x2 + y2 + z2 ≤ 16 A) 666π B) 56,232π C) 56232 5 π D) 1400π Use Stokesʹ Theorem to calculate S (curl F) ∙ n dS∫∫ 8) F = 5yi - 6xj + 2z3k ; C: the portion of the plane 6x + 7y + 4z = 6 in the first quadrant A) - 33 7 B) 0 C) -11 D) 33 7 Use Stokesʹ Theorem to calculate S (curl F) ∙ n dS∫∫ . 9) F = 5yi + 7xj + z3k; C: the counterclockwise path around the perimeter of the triangle in the x-y plane formed from the x-axis, y-axis , and the line y = 5 - 2x (Hint: n = k) A) 25 2 B) 5 2 C) 25 D) - 25 Use Stokesʹs Theorem to calculate S (curl F) ∙ n dS∫∫ . 10) F = (8 - y)i + (5 + x)j + z2k; S is the upper hemisphere of x2 + y2 + z2 = 16 A) 32π B) 64π C) -4π D) 2π