Find the work done by the force F = 4xyi + (2x² – 3xy)j by pushing a particle along the line segment from (-3, -2) to (1, 0) and the arc of the circle in the first quadrant from (1,0) to (0,1).

Q: A force F = -3i +j+6k acts through the point (5,-2,-1). Find its moment about the point (3,0,2).

A: Moment of force about the point is given by cross product of position vector r about a point and…

Q: A yacht sets off from the port in Abu Dhabi and travels 350 km at 145' towards the Persian Gulf and…

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Q: 3. Consider the following helix with vector equation 7 (t) = 2 cos (t) 7 (t) 7 + tỉ + 2 sin (a)…

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Q: Evaluate √(2² + yz sin(xyz))dx+(y²+xz sin(xyz))dy+(x+xysin(xyz))dz where C is the curve following…

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Q: particle started at A(1,0) circled the origin once  an d returned toA(1,0) .what were the change in…

A: Given: A particle started at A(1,0) circled the origin once  and  returns to A(1,0)

Q: The position vector for a particle moving on a helix is c(t) = (5 cos(t), 5 sin(t), t²) . Find the…

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Q: The path r(t) = (t- sin t) i+ (1- cos t) j describes motion on the cycloid x=t- sin t, y=1- cos t.…

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Q: A particle is moving along the curve having the vector equation R(t) = 4 cos ti + 4 sin tj + 6tk.…

A: Given Position vector:- Rt=4costi+4sintj+6tk

Q: Let C be the curve defined by the following system of equations: √x+6y²-2²=0 3y = z

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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: