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- Suppose Vf(x, y) = 6y sin(xy)i + 6x sin(xy)j, F = V f(x, y), and C is the segment of the parabola y = 5x? from the point (1, 5) to (6, 180). Then F. dr =Consider a rectangular surface of length L and width W in the xy plane with its center at the origin: Which of the following are valid expressions for the area vector of this surface? Check all that apply. O (0,0, LW) O (W, L, 0) O (0,0, -LW) O (LW, LW, 0) O (0, LW, 0) O (L, W, 0)b) Given F(x, y, z) = (x³ + cosh z) i+ (2y³ – 3r²y)j – (x² + 4y²z) k. Use Gauss's theorem to calculate //F .n dS where n is the outward unit normal of o, the surface bounded by the planes, x = 0, z = 0 and x + z = 6, and the parabolic cylinder x = 4 – y².
- The surface integral of V = x + yŷ + (z - 3)2 over the sphere of unit radius centred at the origin is Select one: ○ a. 4π ○ b. O ○ c. 4π/3 ○ d. πy 9 4 Let C₁ be the ellipse + = 1 traced counterclockwise and C₂ be the paral- lelogram with vertices at (2, 1), (1,−1), (-1,−1), and (0, 1) traced clockwise. Define C = C₁ C₂ F(x, y) = (1 − y, x + c²²). [F-dR by direct computation. (a) (b) (c) Evaluate Use Green's Theorem to find the value of Use (a) and (b) to find the value of Jo C₂ [.F. dR. F.dR.A triangle in the xy plane is defined with corners at (x, y) = (0,0), (0, 2) and (4, 2). We want to integrate some function f(x, y) over the interior of this triangle. Choosing dx as the inner integral, the required expression to integrate is given by: Select one: o Sro S-o f(x, y) dx dy x=0 2y y=0 O S-o So F(x, y) dæ dy O o S f(x, y) dy dæ O So So F(x, y) dx dy x/2 =0
- 1. Let C denote the positively oriented boundary of the square whose sides lie along the lines x = +2 and y = +2. Evaluate each of these integrals: e- dz COS Z z dz dz; (b) z(z2 +8) (a) (c) - (ai/2) (d) cosh z dz: (e) c (z - tan(z/2) dz (-2 < x0 < 2). Ans. (a) 2n; (b) ni/4; (c) -ni/2; (d) 0; (e) in sec2 (xo/2).Find the point at which the line meets the plane. x = 1+ 6t, y = - 2+ 2t, z= - 2+ 5t; x+ y+z= - 3 The point is (x,y,z) = (Type an ordered triple.)Considering the following points in rectangular coordinates: P, (0,0,2) and P2(0, v2, v2), determine the line integral of F = ra, + 2ra, + 3a, from P, to P2.
- 2 points in the xy plane have A = (2, -4) and B = (-3,3). Determine the distance between points A and B and find their polar coordinates.Find the line integral of F(x, y, z) = x°i + y°j+ zk along the line segment C from the origin to the point (4, 2, 6). F. dr =Let 5 = (-1m)î + (2m)ĵ – (3m)k and T = (4m)î – (6m)k. The angle between these two vectors is given by: Select one: cos 1(14/15) O b. cos 1(14/27) O a. cannot be found since S and T do not lie in the same plane O d. cos (11/15) cos '(104/225) Oe.