QUESTION 2 Let C be the oriented curve parameterized by r(t)=(t,t> for 0st s2 (so C is actually an oriented line segment). Let F(x,y)= (F1F2) where F1=4x and F2=xy.Compute the vector liné integral dr. QUESTION 3

Q: vector-valued function 7(t) the parametric equations of the line.) b) Find the distance of the point…

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Q: 5. (a) (b) The surfaces (x - 2)² + z² = 9 and y = z² intersect along the curve C. Find a vector…

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Q: Develop vector and parametric equations for the line through (7,2,-1) and parallel to r = (4,7,0) +…

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Q: Calculate the gradient vector Vf of the function f(x, y) = x² – x + y- x²y – 2y2 at the point (2,1)…

A: According to the given information, it is required to determine the gradient vector of the given…

Q: Sketch the parametric curve of r(t) =.Then, find and draw the tangent vector r'(0)

A: 2cos t, 2sin t,3To Find:a. Sketch the parametric curve.b. Find and draw the tangent vector r'(0).

Q: Represent the line segment from P to Q by a vector-valued function. P(4, 0, 7), Q(3, -3, 4) r(t) =…

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Q: 4. Find upper and lower bounds for the sequence 3n+7 n n=1

A: To find the upper and lower bounds for the sequence .Concepts used:Let  be a sequence. Then a number…

Q: equations

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Q: 2) Sketch the curve with the vector equation r(t) = costi- costj+ sin tk. Show the direction of…

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Q: 3xyz = 63 and 3x² + 4y² + 8z² = 423 at point (−3, 1, −7). ř(t) = with -∞ < t <∞

A: The parametric form of a line that passes through x0, y0, z0 and has the direction ratios: a, b, c…

Q: At which point P of the curve defined vectorially with r (t) = (21, 4t + 1, t) is the tangent vector…

A: The general form of the parametric equation of the curve in three dimensional space is given by…

Q: 3. Turn in: Find the equation for the tangent line to the curve defined by the vector-valued…

A: We have to find the tangent line For this we doing derivative.The procedure is given below

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: