3. Show that the line integral + 2xy) dx + x² dy + 2xz dz, is independent of path in R3, where C runs from (2, 1,3) to (4, -1,0). Evaluate this line integral.

Q: af 11. Suppose + = 0, prove that the line integral | F. fe -dx + ду dr = -dy is inde- pendent of…

A: Here F = fy i+fxj

Q: 1. What should be A for the path independent of (2,4) ry + A 2Ay – 1 dy y? dr+ (1,2) Moreover…

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Q: Evaluate the line integral along the given path: 1. So ry ds, where C : r(t) = 4ti + 3t²j, 0 < t <…

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Q: 2. Evaluate the given line integral, | (-y,x – y) · dR, where C is the line segment from (1,0) to…

A: To find the line integral we use the equation of line segment in parametric form using given points.

Q: Evaluate the line integral along the path C given by x = 2t, y = 4t, where 0 <ts 1. (y - x) dx +…

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Q: Evaluate the line integral §. xy²dx + 2x²y dy over triangular path with vertices (0,0). (2,2) and…

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Q: Evaluate the line integral along the path C given by x = 4t, y = 8t, where 0 <ts 1. / (x+ 3y²) dy #4

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Q: Evaluate the line integral along the path C given by x = 2t, y = 4t, where o sts 1. (y – x) dx +…

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Q: Use Green's Theorem to evaluate the line integral |(y - x)dx + (2x – y)dy for the given path. C: x =…

A: Let's find.

Q: 3. Calculate the line integral S F - d7, where F = (x²y³, x³y?) and C is the path given by C 7(t) =…

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Q: Evaluate the line integral along the path C given by x = 2t, y = 4t, where 0 ≤ t ≤ 1. √(²x- (y - x)…

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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: Evaluate the integral (6x-y)dx + (x+6y)dy along the path C, defined as y = 16-x² from (0, 4) (4, 0).…

A: Answer: The solution is given below:

Q: What should be i for the path independent of (24) ´ry + A 21y – 1 dy (12) over evaluate the integral…

A:

Q: Evaluate the line integral §. xy²dx + 2x²y dy over triangular path with vertices (0,0), (2,2) and…

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Q: What should be A for the path independent of (2,4) 2Ay – x dy. y? XY + A (1,2) over evaluate the…

A: Consider the provide integral,

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

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Q: Evaluate the line integral fe y dx + x dy where C is the parameterized path x = , y = f, 3 <iS9.…

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Q: Show that in the following problems, the line integrals are independent of path of integration and…

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Q: 2. Evaluate the given line integral, (x – y, y) · dR, where C is the line segment from (0, 1) to…

A:

Q: Evaluate the line integral along the path C given by x = 2t, y = 4t, where 0 ≤ t ≤ 1. [(x + 3x²) dy

A: To evaluate: ∫C(x+3y2)dy...(1) where C is given by x=2t, y=4t⇒dy=4dt and 0≤t≤1

Q: Use Green's Theorem to evaluate the line integral (y - x)dx + (2x – y)dy for the given path. C: x =…

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Q: Find the line integral of F =9xy*ï +(x² + 3),yj over the path L.

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Q: Evaluate the line integral along the path C given by x = 2t, y = 4t, where 0<ts 1. (y – x) dx +…

A: Just I have used the given parametric representation of the curve C to change the given line…

Q: Evaluate the line integral along the path C given by x = 2t, y = 6t, where 0 s ts 1. (y – x) dx +…

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Q: Evaluate the line integral cy dx + x dy where C is the parameterized path x = y = t° , 2<t< 6. Scydx…

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Q: Use Green's Theorem to evaluate the line integral (y – x)dx + (2x – y)dy for the given path. C: x =…

A: Given integral is ∫Cy−xdx+2x−ydy where C:x=2cosθ,y=sinθ,0≤θ≤2π. Green theorem is given by…

Q: Evaluate the line integral along the path C given by x = 2t, y = 6t, where 0 ≤ t ≤ 1. √((x-x) (y -…

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Q: 2. Evaluate the given line integral, (-y,x – y) · dR, where C is the line segment from (1,0) to…

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Q: Evaluate the line integralſ (x + 2y) dx + x² dy, where C consists of line segments from (0,0) to (2,…

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Q: Evaluate the line integral T, (“+y) dx + 4 ry dy where L is the path, -> r) = (3 t, 2) with 0 <t <1.…

A: We evaluate given integral

Q: Evaluate the line integral di the path over C from to Q(2,4) P(O,) where F(xy) 2xy°2+3y5 13D to ネソーメ…

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Q: 15. Evaluate the line integral J.(3xy* - 6xy ]dx + (4x°y³ - 3x?) dy, where C is any path from (1,2)…

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Q: Use Green's Theorem to evaluate the line integral (y - x)dx + (2x - y)dy for the given path. C: x =…

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Q: Compute the path integral of F = ⟨ y , x ⟩ along the line segment starting at    ( 1 , 0 ) and…

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Q: 4. Find path integral J xdx+ ydy, if L is the segment: x-1,y= and re (0.1]

A: Given x=t, y=t2, t∈0,1

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

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Q: Evaluate the line integral IF.då over the path C to Q (2,4) from P(0,) where F(xy) = 2xy2 + 3y f 131…

A: According to question, Given function, F→x, y = 2xy2 i^ +3y j^,We have to calculate line integral…

Q: Evaluate the line integral ∮C x dy−y dx where C is a circular, closed path defined by x^2+y^2=16

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Q: Find the line integral along the given path C. Ju- |(x- y)dx, where C: x = t, y = 5t + 6, for 0 sts2…

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Q: 1. What should be A for the path independent of (2,4) ry +A -dr dy (1,2) Moreover evaluate the…

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Q: 11 Calculate the line integral f Gy)ds for the given junc rion and path xiyl=xy KHt)=<3+,2t=-37 for

A: r(t)=&lt;3t,2t-3&gt;=&lt;0,-3&gt;+t&lt;3,2&gt; x(t)=3t  and y(t)=2t-3 ds=32+22dt=13dt 0≤t≤1

Q: 5, find the line integrals along the given path C. In 13. | (x – y) dx, where C: x = 1, y = 21 + 1,…

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Q: 4. Evaluate the line integral along the path C given by x = 2t, y = 10t, where 0 ≤ t ≤ 1. I (a) √(x…

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Q: Use Green's Theorem to evaluate the line integral (y – x)dx + (2x - y)dy for the given path. C: x =…

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Q: O Evaluate the line integral over The Cur ve C. SCx+y+2)dx +6x +y-2)dy+ (x-y+2)dz where c is the…

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Q: Evaluate the given line integral. | (y, -x) dR, where C consists of 4y = x² from (-2, 1) to (2, 1)…

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