1 + f 2 x + f 2 y dy dx, where f(x, y) = 2 + 3x + 4y, is

Transcription

1 1. The value of the double integral (a) (b) 15 8 (c) 75 (d) f 2 x + f 2 y dy dx, where f(x, y) = 2 + 3x + 4y, is 2. What is the value of the double integral interchange the order of integration.) π/ x cos(xy) dx dy? (Hint: You may need to (a) π 2 (b) 0 (c) 2 π (d) 2 3π 3. The volume of the solid lying under the plane x 2y + z = 1 and above the region in the xy-plane bounded by the lines y = 0, x = 1 and the parabola y = x 2 is (a) (b) 1 5 (c) (d)

2 4. After changing to polar coordinates, the integral D xy 2 da, where D is the region in the first quadrant enclosed by the circle x 2 + y 2 = 4 and the lines y = 0 and y = x, evaluates to (a) (b) (c) (d) If the density function is ρ(x, y) = 1 + 3x + y, the mass of a triangular lamina with vertices (0, 0), (1, 0) and (0, 2) is (a) 2 3 (b) 8 3 (c) 0 (d) Find the value of xy dv, if T is the solid tetrahedron formed by the planes y = 0, z = 0, T x = 1 and x y = z. (This is the tetrahedron with the vertices (0, 0, 0), (1, 0, 0), (1, 1, 0) and (1, 0, 1)). (a) 1 2 (b) 1 72 (c) 1 6 (d)

3 7. If the cylindrical coordinates of a point in space are (3, π 2, 2), its rectangular coordinates must be (a) (3, 0, 2) (b) (0, 3, 2) (c) (0, 3, 2) (d) ( 3, 0, 2) 8. If E is the region between the spheres x 2 + y 2 + z 2 = 4 and x 2 + y 2 + z 2 = 9, it can be described in spherical coordinates by (a) E = {(ρ, θ, φ) 2 ρ 3, 0 θ 2π, 0 φ π} (b) E = {(ρ, θ, φ) 0 ρ 3, 0 θ 2π, 0 φ π/2} (c) E = {(ρ, θ, φ) 0 ρ 2, 0 θ 2π, 0 φ π} (d) E = {(ρ, θ, φ) 2 ρ 3, 0 θ 2π, 0 φ 2π} 9. Evaluate the integral R (x+y)ex2 y 2 da, where R is the rectangle enclosed by the lines x y = 0, x y = 2, x+y = 0 and x+y = 3. (Hint: Use change of variables. Note that x 2 y 2 = (x y)(x+y). Do not forget to take care of the Jacobian.) (a) e3 1 4 (b) e (c) 2e 5 7 (d) e

4 10. In spherical coordinates, the equation of a surface is φ = π/6. Then the surface must be a (a) cone (b) sphere (c) hemisphere (d) paraboloid 11. The integral ( t 2 i sec 2 t j + 1 ) 1 + t 2 k dt evaluates to (a) t3 3 i tan t j + tan 1 t k + C (b) 2t i 2 sec 2 2t t tan t j (1 + t 2 ) 2 k + C (c) t 2 i sec 2 t j t 2 k (d) t3 + t i (t + tan t) j + (t + tan 1 t) k + C 3 (Here C is the vector constant of integration.) 12. The velocity vector of a particle that has the acceleration vector as a(t) = 2i + 6t j + 12t 2 k and the initial velocity as v(0) = i is (a) v(t) = 2t i + 3t 2 j + 4t 3 k (b) v(t) = (t 2 + t)i + (t 3 + 1) j + (t 4 1) k (c) v(t) = (2t + 1)i + 3t 2 j + 4t 3 k (d) v(t) = (t 2 + t)i + t 3 j + t 4 k 4

5 13. A particle moves with position function r(t) = t 2 i + t 2 j + t 3 k. Then its tangential component of acceleration (given by r (t) r (t) ) is r (t) (a) (b) (c) (d) 6 2t 2 8t2 + 9t 4 8t + 18t 3 8t2 + 9t 4 18t + 8t 3 9t2 + 8t 4 6 2t 2 9t2 + 8t Find the length of the curve γ(t) = 12t, 8t 3/2, 3t 2, 0 t 1. (a) 9 (b) 12 (c) 15 (d) Find the curvature of the curve γ(t) = e t, t, t 2 at t = 0. (a) 1 (b) 3/2 (c) 9/4 (d) 9/8 5

6 16. Find an equation of the tangent plane to the parametric surface γ(u, v) = u 2 + 1, v 3 + 1, u + v at the point (5, 2, 3). (a) 3x + 4y 12z + 13 = 0 (b) 3x 4y 12z + 13 = 0 (c) 3x + 4y + 12z + 13 = 0 (d) 3x + 4y 12z 13 = Find the area of the part of the plane x + 2y + 2z = 4 that lies in the first octant. (a) 3 (b) 6 (c) 9 (d) Find the volume of the tetrahedron bounded by the planes x + 4y + 12z = 12, x = 0, y = 0, and z = 0. (a) 3 (b) 6 (c) 9 (d) 12 6

7 19. Evaluate the surface integral S F d S for the vector field F (x, y, z) = xz, x, y, where S is the hemisphere x 2 + y 2 + z 2 = 16, y 0, oriented in the direction of the positive y-axis. (a) π/2 (b) π (c) 3π/2 (d) 2π 20. Evaluate the surface integral S 3x ds, where S is the graph of y = x2 + 63z over the rectangle 0 x 3, 0 z 2. (a) 122 (b) 244 (c) 366 (d) Find a potential function for the conservative vector field F (x, y, z) = 3y 2 z 3 + 2xy 2, 6xyz 3 + 2x 2 y 2z 2, 9xy 2 z 2 4yz. (a) xy 2 z 3 + x 2 y 2 2yz 2 (b) 3xyz 3 + x 2 y 2 2yz 2 (c) 3xy 2 z 3 + xy 2 2yz 2 (d) 3xy 2 z 3 + x 2 y 2 2yz 2 7

8 22. Evaluate the line integral C P dx + Qdy, where P (x, y) = 4xy, Q(x, y) = x2 + e y2, and C is the three edges of the rectangle with vertices (0, 0), (3, 0), (3, 1), and (0, 1) starting from (3, 0) to (3, 1) to (0, 1) to (0, 0). (a) 6 (b) 6 (c) 9 (d) Evaluate the line integral F d γ, where F (x, y) = C counterclockwise orientation. (a) 0 (b) π (c) 2π (d) 2π y, x x 2 +y 2, and C is the circle x 2 + y 2 = 1 with 24. Evaluate the line integral C F d γ, where F (x, y) = f and f(x, y, z) = 3x 2 z 5 x y 4 and C is the helix given by x = cos t, y = sin t, z = t 2π, 0 t 2π. (a) 0 (b) 1 (c) 2 (d) 3 8

9 25. Evaluate the integral y cos z ds, where C is the helix given by x = cos t, y = sin t, z = t, C 0 t π/2. (a) 0 (b) 1 (c) 2/2 (d) Find the Curl and divergence of the vector field F (x, y, z) = y + z + x 2, x + z + y 2, y x + z 2. (a) CurlF = 0, 2, 0 and divf = 2(x + y + z) (b) CurlF = 0 and divf = 2(x + y + z) (c) CurlF = 0, 2, 0 and divf = x + y + z (d) CurlF = 0 and divf = x + y + z 27. The radius r and the center O of the sphere defined by x 2 6x + y 2 + z 2 2z = 0 are (a) r = 10, O = (3, 0, 1) (b) r = 10, O = (3, 0, 1) (c) r = 10, O = ( 3, 0, 1) (d) r = 10, O = (0, 0, 0) 9

10 28. The distance D between the point P = (3, 0, 1) and the plane with equation x + y + 2z 1 = 0 is equal to (a) D = 1 6 (b) D = 6 (c) D = 4 6 (d) D = 2π Two vectors a and b are parallel if and only if (a) a b = 0 (b) a b = 0 (c) a b = 1 (d) a = ± b 30. An equation of the plane that passes through the points P = (1, 3, 2), Q = (3, 1, 6) and R = (5, 2, 0) is (a) x + y + z = 6 (b) 6x 10y 7z = 0 (c) 6x + 10y + 7z = 50 (d) 6x + y + 7z = 7 10

11 31. The direction angles of the vector a = 1, 1, 0 are (a) α = π 4, β = 3π 4, γ = π 2 (b) α = π 4, β = 3π 4, γ = 0 (c) α = π 2, β = π 2, γ = π (d) α = π 2, β = π 2, γ = The domain D and the range R of the function f (x, y) = 9 x 2 y 2 are (a) D is the square { (x, y) R 2, 0 x 3, 0 y 3 } and R = [0, + ) (b) D is the disk centered at (0, 0) with radius 9 and R = [0, + ) (c) D is the disk centered at (0, 0) with radius 3 and R = (, + ) (d) D is the disk centered at (0, 0) with radius 3 and R = [0, + ) 33. The non-empty level curves of the function f (x, y) = 4x are (a) concentric circles centered at (1, 1) (b) parallel lines (c) squares with center (0, 0) (d) ellipses 11

12 34. The following limit (a) is equal to 1 lim (x,y) (0,0) xy x 2 + y 2 (b) is equal to 0 (c) does not exist (d) is equal to The tangent plane to the elliptic paraboloid z = x 2 + 4y 2 at the point (x 0, y 0, z 0 ) = (1, 1, 5) is equal to (a) z = 2x + 8y + 5 (b) z = 2x + 8y 5 (c) z = x + 4y (d) z = 2x + 8y 36. If f (x, y) = x 2 y + 3xy 3 and x = sin (t) and y = cos (t), then the value of df dt (a) 3 (b) 0 (c) 3 (d) π at t = 0 is equal to 12