Let C: r(t), te [a,b], be a regular curve in R³, oriented from the point P = r(a), up to the F-dr can be calculated with the fundamental point Q=r(b). If x,y,z #0, the integral c theorem of the line integrals if we take: A) F(x, y, z)=(, +3y², -32²) B) F(x, y, z) = (+3y², -32²) C) F(x, y, z) = (+3y², -32²) D) F(x, y, z)=(+3y², -3y²)

Q: Evaluate the line integral -ydx + xdy along the curve C C : y? = 24x from (24, 24) to (0,0). NOTE:…

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Q: Suppose an object moves in space with the position functionr(t) = ⟨x(t), y(t), z(t)⟩ . Write the…

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Q: (20 points) Evaluate the following line integral along the positively oriented curve C consisting of…

A: To find the final answer for the line integral, we need to evaluate the integral over each segment…

Q: – x²y? x2 + y? f(x, y)

A: fx, y=-x2y2x2+y2 Now, z0=fa, b=f1, -2=-12-2212+-22=-45 Now partially differentiating fx,…

Q: The line integral P F.d7 can be evaluated using Stoke's Theorem where F = (2x, -3, 1) and C is an…

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Q: 3. Let f(z) = 1-z and y(t) = t+it for t E-1,1). (a) Sketch the image of the curve y. (b) Evaluate…

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Q: E Use Green's Theorem to evaluate the line integral along the given positively oriented curve. [ 4³…

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Q: use the Fundamental Theorem of line Integrals to evaluate F. dR, where F (*.y) = < y. - X and C is…

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Q: Use the Fundamental Theorem of Line Integrals to compute the work done F on an object that is moved…

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Q: 5) Evaluate the integral (e* - y) dx + (sin y + x)dy using Green's theorem where Cis bounded by y =…

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Q: Let C be the curve represented by the equations X = 11t, y = t², (0 ≤ t ≤ 1). In each part, evaluate…

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Q: Show that the integral is independent of the path, and use the Fundamental Theorem of Line Integrals…

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Q: (2) Find the work done by F over the curve C in the direction of increasing t. Explain why the sign…

A: Question: Calculate the work done by  Fx,y,z=zi^+xj^+yk^ over the curve  C : r¯t=sinti^+costj^+tk^ ,…

Q: Let C be the curve represented by the equations X = 4t, y = t², (0 ≤ t ≤ 1). In each part, evaluate…

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Q: Evaluate the line integral f, F . dr, where C is the curve given by F(t) = 3 cos ti+ 3 sin t j, 0<t<…

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Q: Sc F. dr where F(x.y) = (3x² – 4v)i + (27 + cos(,³))j

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Q: Evaluate the line integral (cos x + sin y) ds; C is the line segment from (0, 0) to (x, 67).

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Q: Evaluate the line integral where F y = 2x² from (-1, 2) to (2,8).

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Q: Given: Find (a) div F (b) curl F

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Q: F. dr where F = yzi + xzj +xyk and C is the curve x = t, y = 2t², z = –2t² with 0<t < 1.

A: Given F=yzi+xzj+xyk and r=ti+2t2j-2t2k Now the vector field along the curve can be given by…

Q: Sc xe¹zdS, C is the line segment from (0,0,0) to (1, 2, 3)

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Q: x ds, where C is the portion of the parabola from (0,0) to (1, 1), followed by the line segment from…

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Q: Calculate the line integral ∮C√x2+y2dx +y[xy+ ln(x+√x2+y2)]dy using Green’s theorem. The contour of…

A: As per the question we are given a line integral and we have to use the Green's theorem to evaluate…

Q: Q.2: Let C be the line joining A(-1,0), B(0,1), D(1,1), and E(2,0). Evaluate: Scy dz tor and radius…

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Q: Consider the curve C = {(x, y) = R² | y = 25 + ², - 12 ≤ x ≤ 12} in R2. Evaluate the line integral I…

A: Given curve C,Now,