F.dr. QUESTION 12 Let C be the oriented curve parameterized by r(t)=(t,t> for 0sts2 (so C is actually an oriented line segment). Let F(x.y)=(F1F2) where F1=8xand F2=xy.Compute the vector line integral

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A: rt=et,e2t,t0=0

Q: Find a unit vector u that is normal to the level curve of f(x, y) = 5x²y through P = (x, y) = (1,-4)…

A:

Q: What is the unit tangent vector to the curve r(t)=⟨cost, sint, et⟩ at the point where t=0?

A: Given:          r(t)=cost,sint,et at t=0           r→(0)=cos(0),sin(0),e0          r→(0)=1,0,1…

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A:

Q: Eliminate the and write an equation of- carve in part (a) Sobely in cartesian coordinates x, y (e)…

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A:

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A:

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A:

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A: The given vector valued function

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A:

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A:

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: