Evaluate x*zdx – yx dy + 6 dz along C the curve C shown in the figure. A Z 2 2 y 2 x.

Q: Find a parameterization for the curve shown in the figure below. |--1 = 110 -1 r(t) where 0 ≤t≤ 1…

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Q: Evaluate Se (5xy + x² + y²) dx + (x² - y)dy by using Green's theorem, where C is a closed curve that…

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Q: Let C be the curve y = x³ – 3x2 + 4x - 1 joining the points (1, 1) and (2, 3). Find the value of |…

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Q: Evaluate + 2x + 1)dx+ (2xy+4y- 1)dy along the given path C from (0,0) to (1,1). 5. The straight-line…

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Q: Let f (z)=z. Find zp( z) 4} from z =0 to z=4+2 along the curve C given by z =t? +it. (A) -10 - (C)…

A: Use the definition of path integration of analytic function.

Q: Find 2 for the curve dx x²y³ – 2xy = 6x² + 2y3 + 1

A: Topic = Derivative

Q: %3D Let C be the curve connecting (1,1) to (-1,-1) clockwise around the circle x2 + y = 2 -dy Jc x²…

A: Differentiate the given curve w.r.t x 2x+2ydydx=0dydx=-xy Now,…

Q: Let C be the curve y = 8/a for 2.1 < x < 3.8.

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Q: If C is the curve |z| = 3, then cos(z) -dz .3 O - Tİ Ο 2πί 1 O none of these

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Q: Let C be the curve y = x³ – 3x? + 4x – 1 joining the points (1, 1) and (2, 3). Find the value of…

A: Solving the Cauchy integral using Cauchy theorem

Q: Evaluate c (xy + x+z) ds along the curve r(t) = 2ti + tj + (6 – 21)k, 0 sts1. [c kxy +x +z) ds =…

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Q: Evaluate c (xy+y+z) ds along the curve r(t) = 2ti + tj + (2- 2t)k, 0Sts1. (xy+y+z) ds=| (Simplify…

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Q: Which of the following gives the length of the curve y x2 + 2x from x = 0 to x = 5 ? (x2 + 2x) dæ I…

A: Length of the given curve will be

Q: II. Find the length of the following curves. (a) x = 2t , y = cos 4t, 0 <t <

A: Given:    x = 2t    y = cos4t    0 ≤ t ≤ π/2

Q: (b) The curve y = x +3x, from x 1 to x = 3 (c) The ,2 1 from 2 to g

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Q: D A Find thepolar eavation for the cartesian curve (x²ty) - A)r-4 cos o =4x. Br°=4 cos e c)r?=4cos o…

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Q: Express the curve by an equation in x and y given x(t) = e' and y(t) = 7 – e2'. a) O-x2 + y = 7, x 2…

A: simplify the equation and choose correct equation.

Q: 6. Find dyldx and d 2yldx 2. For which values of t is the curve concave upward? x = t 3 - 48t, y = t…

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Q: The value of le ydx + zdy + xdz where C is the curve x = y = z = 2t2 from t = 0 to t = 1 is A В 15 C…

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Q: Q5. The curve y = ax? + bx + c passes through the point (1, 2) and is tangent to the line y =x at…

A: # we are entitled to solve one question at a time, please resubmit the other question if you wish to…

Q: Let f (z)=7. Find f (z )dz from z =0 to z =4+2i along the curve C given by z =t? +it. (A) –10 – (C)…

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Q: a)Find the exact length L of the curve 3y? = (4x – 3)°, 1 0. Answer: b) Evaluate Sxe*dx Answer: 1 c)…

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Q: Let the equation of curve c be x=2t, y=t3, t∈[0,2]. What is the value of the integral ∫c -y2dx +…

A: Explanation of the answer is as follows

Q: Find F-dX where C is the curve X(r) = r°I+( 2r+ 1J, 0<I55, and Fxy) = rI+yJ.

A: 9

Q: dy for the curve defined by x° – y = 2xy? dx Which of the following is O 2x + 3y? 3x2 – 2y O 3x2 –…

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Q: Find the condition that the line x cos a + y ain a =p may touch the curve yh =1. am bm

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Q: The curve r+y-6x2y 0 is symmetric about Oy = 0 Ox = 0 Oy =x None of these

A: We need to check the symmetry of the curve x4+y3-6x2y=0

Q: Evaluate +X+Z) ds along the curve r(t) = 2ti + tj + (3 - 21)k, 0<t<1. cy +x+2) (Simplify your…

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Q: 8) Consider the oriented curve C given by C=C₁U C2, where C₁: y = x³, and C₂: y = x², as shown in…

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Q: Find the work done by F in moving a particle once counterclockwise around the given curve. F= (4x-…

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Q: y = e*, 0<z<1.

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Q: Let y be the part of the curve y = x² running from (-1,1) to (2,4). Calculate xy dx + (x + y) dy.

A: (1)

Q: 5. Let C be the curve parameterized by r(t) = (t,-t,t²) Evaluate y ds.

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Q: 1. Sketch each curve in the z-plane, and sketch its image under w = 2². (a) |z1|=1 (b) x = 1 (e) y²…

A: We are authorized to answer three subparts at a time since you have not mentioned which part you are…

Q: Find the length of the curve r(t) = (1,t2, t³). 0<t < 1.

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Q: Evaluate G(x, y)dx G(x, y)dy and G(x, y)ds on the curve C indicated by = 2t, y = t2, where t is on…

A: If ∫cf(x,y) dx typeof integral is given then the function is integrated with respect to x and…

Q: Draw direction field of the DEQ below, by hand, and sketch an approximate solution curve that passes…

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Q: O Find the length of the following curves 4/2 3/2 + Dx, 0<I<1, 3 i) y(r) = 21 ii) 1 = ² + 21, y=…

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Q: Find the coordinates of the point(s) on the curve 2y = 10 – a² that are closest to the origin. а) О…

A: Given that

Q: The length of the curve defined by x =(v² + 2)E, y € [0,1]. Select one: none O 12

A: Arc length of the curve x=g(y), y∈[a, b] is, S=∫ab1+(g'(y))2 dy

Q: Let C: [o,2]> R be a curve defined by C(t) = (t²,t'). Evaluate. Sy dx x dy.

A: Given:  C: [0, 2]→ℝ2   is a curve defined by                C(t)=(t2, t3)implies x=t2 and…

Q: Find the length of the curve r(t) = (√2t)i + (√2t)j + (1 - t2)k from (0, 0, 1) to (√2, √2, 0)

A: given, r(t)=(2t)i +2tj+1-t2kwe have to find the length of the  given curve from (0,0,1) to 2,2,0

Q: Let C be any curve.

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Q: IF=(x+y)i+(y– )j+z?k over the curve r(t) = (t²,t²,t*), 0sts1

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Q: Evaluate | yd x - x dy along the curve C shown in the figure. (a) (0,3) (3 ,0)

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Q: Let Q = 4x? + 6xy+ 4y². The points on the curve Q = 7 which are closest to the origin are:

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Q: y(z) along the curve given by r" = e*.

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Q: py*(1 + x*)dx + 2ytan xdy ,where C is closed curve xi +yi = 1

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Q: Sc xydx + (x + y)dy Calculate on the y=x? curve from (-1,1) to (2,4) wwww wwm wwww

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Q: Let C be the curve connecting (1,1) to (-1,-1) clockwise around the circle x2 + y = 2 1 -dy + y? + 1…

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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: