Prove that fe, r*dx + rydy = 0 where Co is the line segments from (0, 1) to (0,0) by using parametrization.

Q: Evaluate the line integral / Vxy° +3 dx +6xy" dy, where C any curve joining (1, -1) to (16, 1).

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Q: Evaluate the line integral -y dx + z dy + 2x dz, where C is the helix by 7(t) = (cos t, sin t, t),…

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Q: Evaluate the line integral, where C is the given curve. | xyz? ds, C is the line segment from (-1,…

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Q: Evaluate the line integral using the Fundamental Theorem of Line Integrals. Use a comp cos(x) sin(y)…

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Q: Suppose F(x, y) = -yi + xj and C is the line segment from point P = Q = (0,4). (5,0) to (a) Find a…

A: The solution are next step

Q: Apply runge kutta method of order two on du/dx = sin(xu), u•=0.1,h=0.1 to find u(0,3)

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Q: Let C be the line segment from [1 – i,2+ 2i). (a) Parametrize C. (b) Integrate 2 over C.

A: According to the given information, Let C be the line segments from [1-I,2+2i]. For part (b) it is…

Q: Evaluate the line integral, where C is the given curve. | xyz? ds, C is the line segment from (-2,…

A: Find the equation of line in parametric form.

Q: Compute the line integral (х — y) dx + xe x dy, where C is the closed curve consisting of two line…

A: line integration

Q: Evaluate the line integral / V(ry + xz + yz) · dĩ over the curve F(t) = (t, 2t, 3t) for 0 <t < 4…

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Q: State two other forms for the line integral ∫C F ⋅ n ds given that F = ⟨ƒ, g, h⟩ .

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Q: Use Green's Theorem to calculate line integral o sin(x²) dx + (3x – y) dy where C is a right…

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Q: 2. Evaluate directly the line integral fo a dy where C is the quarter-circle from (0, 4) to (4, 0)…

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Q: Evaluate the line integral | F•ds where F = 2xyi + (x – y²)j and C is (a) y = x from (0,0) to (1,1)…

A: For the solution follow the next steps.

Q: Evaluate the line integral ,F. dr where F = (sin x, –2 cos y, 5xz) and C is the path given %3D by…

A: Integral F.dr = integral F(r(t)).r'(t) dt

Q: Use Green's Theorem to evaluate the line integral S(x² + y²)dx + (x² – y²) dy where C is the…

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

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Q: Evaluate the line integral Fdr where F = 4.xyi – 8yj+ 3k and C is a curve given by y = 2x, z = 3…

A: Answer

Q: Use Green's theorem in order to compute the line integral (2sin z + 3y²) dx + (sin(4®) + 2x³) dy…

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Q: y*dx + x*dy = (N, – M,) • dA (м, - м,) ал

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Q: Evaluate the unoriented line integral (2æ² + y² + z²) ds along the parametric curve a : [0, 1] → R³…

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

A: Line integral will be path independent if vector field whose integral we need  is conservative…

Q: Suppose F(x, y) = -y sin(x)i + cos(x)j. Find a vector parametric equation for the parabola y = x²…

A: The solution is following.

Q: Evaluate the line integral along the curve C. (y + z)ds, C is the straight-line segment x= 0, y =…

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Q: | (xy dx + 2y dy)

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Q: Use Green's Theorem to evaluate the line integral. y dx + 7x dy C: square with vertices (0, 0), (0,…

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Q: Evaluate the line integral, where C is the given curve. | xyz? ds, C is the line segment from (-3,…

A: Integration

Q: Evaluate the line integral 2xyz dx + x²z dy + x?y dz where C is an oriented simple curve connecting…

A: Given line integral is,

Q: |[-y dx + xy dy]

A: See the attachment

Q: | 5y? dx +xy dy

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Q: Set up an integral to compute the arc length L of the curve C: r(t) = (, 4t, v2t) %3D from the point…

A: I am going to setup an integral to compute the arc length of the given curve using the required…

Q: Evaluate the line integral Jxy + z)ds where c the arc of circle x + y = 4, z= 0 from (2, 0, 0) to…

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Q: Evaluate the line integral fx ds between (4,0) and (0,4), where C is the circle of radius 4 centred…

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Q: Evaluate the line integral | F•ds where F = 2xyi +(x – y²)j and C is (a) y=x from (0,0) to (1,1) (b)…

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Q: Evaluate the line integral Sp(xi+ yyj) · dr where D is the oriented curve shown below. (1,1) y = x²…

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Q: Find the work done by the force field F(x,y)=5x2i−2xyj on a particle that moves once around the…

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Q: Evaluate the line integral (y + z) ds along the curve C defined by the straight-line segment æ = 0,…

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Q: 7. Use Green's Theorem to compute the line integral | ry? dx – x² dy, where C is the triangle with…

A: ∫xy2dx - x2dy triangle vertices are (0,0), (2,0), and (2,2)

Q: Given the scalar functionf (x, y) = 2xy, calculate the line integral along the ellipse curve+ 1 from…

A: Given: f(x,y)=2xy From the point (3,0) hesta (0,2)

Q: Evaluate the integral ∫(1,2,3)(5,9,−4)y dx+x dy+2 dz by finding parametric equations for the line…

A: To evaluate the integral

Q: Use Green's Theorem to evaluate the line integral cos (y) dx + x²sin (y) dy along CoS the positively…

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Q: Compute the area A under the parametrized curve c(t) = (e',5t) for 0 <t < 1. %3D (Give an exact…

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Q: Evaluate the line integral ſp(xi + Vyi) · dr where D is the oriented curve shown below. (1,1) y = x2…

A: We know that r=xi+yj As, the path followed is y=x2 So, r=xi+x2j Let x=t, then  r=ti+t2jdrdt=i+2tj…

Q: Evaluate the line integral [F-dš where F = 2xyi + (x – y')j and C is (a) y =x from (0,0) to %3D…

A: For the solution follow the next steps.

Q: Find the line integral with respect to arc length (2x + 7y)ds, where C is the line segment in the…

A: Given end points of line segment are  P=8,0 and Q=0,3

Q: (2x + y)dx + (x + 2y)dy

A: Given integral is:  where C is any curve that joins the point (1,1) with the (2,2).

Q: Evaluate line integral F. dr, where F(x, y) =(2xy)i + (x²+3)j, and C is the path given by r(t) = t =…

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Q: please answer the (a) (b) (c)  parts

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Q: 2. Find the parametrization of the tangent line to the space curve r(t) = (-t3, In(t), e² ) at t =…

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Arc Length

Arc length can be thought of as the distance you would travel if you walked along the path of a curve. Arc length is used in a wide range of real applications. We might be interested in knowing how far a rocket travels if it is launched along a parabolic path. Alternatively, if a curve on a map represents a road, we might want to know how far we need to drive to get to our destination. The distance between two points along a curve is known as arc length. 

Line Integral

A line integral is one of the important topics that are discussed in the calculus syllabus. When we have a function that we want to integrate, and we evaluate the function alongside a curve, we define it as a line integral. Evaluation of a function along a curve is very important in mathematics. Usually, by a line integral, we compute the area of the function along the curve. This integral is also known as curvilinear, curve, or path integral in short. If line integrals are to be calculated in the complex plane, then the term contour integral can be used as well.     

Triple Integral

Examples: