Find an equation for the ellipse x2 y² 1 16 25 in parametric form (x(t), y(t)). 1. (4 cos 2t, 5 sin t), 0

Q: 4) Find the critical points of the parametric equations x = t* - 1 y = t3 - t

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Q: 37 <t< y2 = 1 81 The parametric curve x = 9 sin 2t and y = 9 cos 2t, where traces the ellipse 81 O…

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Q: 13.7. Let y be the positively oriented ellipse y? = 1 with a2 – b² = 1. Show that dz + 27, VI- where…

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Q: Question attached in photo

A: To calculate the line integrals with prescribed orientations of the paths

Q: 8. Find the positive constant a(> 0) so that the parametric x = at, y = -t? is a solution to y =…

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Q: d. P = 4x1 – 2x2, Q = 2x1 + 6x2; D is the interior of the ellipse: x1 = 2 cos 0, x2 = = sin 0, 0 E…

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Q: (c) e="(cos(y)dr + sin(y)dy) where C is the square with vertices (1, 1), (1, –1), (-1, –1) and (-1,…

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Q: 4) Find the critical points of the parametric equations x = t* – 1 y = t3 -t

A: Topic:- application of derivatives

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Q: (a) The associated parametric curve is one-half of what familar conic? Sketch the exact curve. r (t)…

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Q: Give a parameterization for the ellipse 16x² + 9 y² = 144 that begins at the point (3,0) and…

A: Parametric equation of the ellipse x2a2+y2b2=1 is given by x(t)=a cosθ, y(t)=b sinθ, t∈[0, 2π]

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Q: 8. Solve for the critical points of the parametric equations x = t3 – 1,y = t² +t %3D

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Q: Solve for the critical points of the parametric equations x = t3 -1, y = t2+t

A: To find the critical points

Q: For the parametric equations: * = 22e2t and y = 3e3* da (*) dt dy (é) dt dy (á) da

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Q: 3. (a) The surface f(x, y, z) = 1+y - 2 = 0 and g(x, y, z) = x+2-4 0 mect in an ellipse E. Find the…

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Q: A cartesian equation of the parametric curve x = 12t + 11 E - 5, y = JE is OAx = (3y – 1 ( 4y +. =…

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Q: 9.) a. Find the parametric equations for the tangent line to the Curve at' the Specific point. X =t°…

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Q: Give a parameterization for the ellipse 9 a? + 36 y? = 324 that begins at the point (6,0) and…

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Q: 17. If y = tx, where t is the parameter, what are the parametric equations of the curve x' + y' = 8?…

A: solution:-y=txx3+y3=8x3+tx3=8x31+t3=8x3=81+t3x=21+t33

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: