Find the work done by the force, F(x,y) = 2xyi+ 4x*y, along the curve C given below in the counter clockwise direction: C, : y = 0, C, : x =1, C,: y x }C=C,uc, uc,

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Q: Find the work done by the force F = 9xy i + 6yj acting along the piecewise-smooth curve consisting…

A: Given that F=9xy i+6y2 j Let r=xi+yj+zk ⇒dr=dx i+dy j+dz k Now,  F.dr=(9xy i+6y2 j+0k)(dx i+dy j+dz…

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Q: Find the work done by the force F(x,y, z) = x*j + 5zk over the curve C: r(t) = cos t i+ sintj+ tk,…

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Q: Compute dS where F = (x, Y, z) and S is the surface of the cone z2 = x² + y² for 0 < z < 4 plus its…

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Q: Find the work done by the force F = xyi + (y - x)j over the straight line from (1,- 2) to (2, - 4).

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Q: What is the value of F.ds, where F = 4xz,- y° i, + yz, ? Here, s is the surface bounded by x = 0, x…

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Q: Find the work done by the force F(x, y) = acting on a point along the smooth curve C, where C is the…

A: The given question is from the topic of Advanced Calculus

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A: Convert function into parametric function. Then find work done by doing line integration.

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A: In this question, we need to find the force exerted by block Y on block X.

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Q: Compute the surface area of the surface segment parametrised by x(u, v) = u² cos v, y(u, v) = u² sin…

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Q: Find the work done under the influence of the force E = 3xyi – 5zj + 10xk along the curve x= t² + 1,…

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Q: Find the work done by the force F 6xy i + line segments from (-4, 4) to (0, 0) and from (0, 0) to…

A: Consider  F(x,y)=M(x,y)i+N(x,y)j is vector valued function then : if ∂M∂y=∂N∂x then F is called to…

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Q: Find the work done by the force F =xyi + (y - x)j over the straight line from (- 2, - 1) to (-1,0).

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Q: F= (e* – y+ xcosh(x))i+ (y³/2 + x)j.

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Q: alculate the work done by the force F (x, y) = ⟨ex, x2y⟩ around the border of the region R = {(x, y)…

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Q: Suppose that an object moves in the positive direction along a coordinate line over the interval [a,…

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Q: Find the surface area of the helicoid (spiral ramp) with vector equation F(u, v) = (u cos v, u sin…

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Q: Find the work done by the force F(x, y) =(x' – y') i+xy' j acting on a point while travelling along…

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Q: Find the flux of F = 5zi+ 3y²j+ 2xk through of the total surface of the closed cone y = Vx² + z²,…

A: Flux of vector F through a surface S  can be given by  flux= ∫∫SF.ds We will use Divergence theorem…

Q: Let I be the moment of inertia of the plate with density p = 1 bounded by y = ez, y = e, x = 0 about…

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Q: Use Stokes' Theorem to calculate fe F-dĩ, where F = zyi+3=j+5zk, and C is the closed path around a…

A: Here given that F→=xy i^+3z j^+5x k^ and curve C in xz-plane. To find: ∫C F→·dr→ by Stokes' theorem.

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A: The solution of line integral is given as follows :

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Q: Find the work done by the force F(x, y, z) = (x, —z, 2y) in moving straight from (0,0,0) to (1,1,0),…

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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: