Do the following tasks using Mathematica. i) Find the length L of the curve y = r+ cos r, 0SST/2 Úse any one of the following formulas dy L= 1+ de de dr. 2 1+ dy dy L =

Q: Find the arclength of the curve z 9 cos(9t), y = 9 sin(9t) with 0 <t< 36

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Q: Find the exact length of the curve. r(t) = (51, 3cos (1), 3sin (1)), -4≤1≤4 OA. 16√3 OB. 0 OC. 8/26…

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Q: Find the length of the following two-dimensional curve. r(t) = (cos t, sin ), for 0sts 6

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Q: Find the length of the following two-dimensional curve. r(t) = (cos t, sin , for 0sts5

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Q: Given the curve r(t) = ,0stsn/2. Calculate the arc lengt O L-3 L=12 L=8 O L=3/2

A: Given curve is r→t=cos3t,sin3t and 0≤t≤π2 We have to find the arc length of the given curve. The…

Q: Find the length of the curve: "(1) =/21i +e'j+e°k, 0 si sl

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Q: Find the length of the curve (t) e cos(t), y(t) = e" sin(t) for 0 <t<8. Give an exact answer,…

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Q: x4 y 1 ¡[1,2] %3D 4 x² +

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Q: ind the length of the curve x cos(t) + In(tan(;t)), y = sin(t), /4<t< 3n/4. %3D ength =

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Q: (B) The length of the curve r(t)=(1+1i+(1-nj+rV2k from the point t= to the point t=; is 2. - 112

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Q: 5. Do the following tasks using Mathematica. i) Find the length L of the curve y = 1 In a, 1<¤< 2…

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Q: Find the arc length of the curve defined by R(t) = 12t î+5costj+(3-5sint)k from t=0 to t=2. O 26 21…

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Q: The arc length of the curve y=3-4x2, where 0<x<1is [1-37/37] O 54 /37-1] 54 1-/37] O, 54 4 137/37-1]

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Q: Find the length of the curve. r(t) = (3t, 3 cos(t), 3 sin(t)), -3 sts 3 %3D

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Q: Find the length of the curve r(t) = (cos 3t, sin 3t, 2t3/2),0 sts 3. %3D

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Q: Find the length of the curve r(t) = e³t cos(t), y(t) = et sin(t) for 0 < t < 10. Give an exact…

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Q: Find the length of the following two-dimensional curve. r(t): = (cos t, sin t) , for 0stsT The…

A: The formula for finding the length of the curve is explained in step 2 with a general observation…

Q: Find the exact length of the curve. x = 7 cos(t) – cos(7t), y = 7 sin(t) – sin(7t),

A: given : formula for arc length is as below :

Q: Find the length of the following two-dimensional curve. (21+1,3/24 r(t) = 2 3 for 0 sts6 L- (Type an…

A: To find The length of the given two dimensional curve.

Q: What is the length of the curve x = cos t + t sin t, y = sin t – t cos t, (0 < t < 2x) ? Lütfen…

A: First I have written the Arclength formula for the parametric curve , then using that formula and…

Q: The arc length of the curve x| + \yl = 1 is Select one: a. 4/2 b. 4 C. 2

A: The graph of the function helps to find the arc length of the curve. Since lengths of all the sides…

Q: Find the exact length of the curve. y = In(sec(x)), OsxSa/4

A: Length of a curve f(x) in interval [a,b] is given by ∫ab1+(f'(x))2dx

Q: Use the arc length formula to find the length of the curve y = 2-3x, -2 is less than or equal to x…

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Q: Consider the curve y = zy+1. What is the value of- dz at (0, 1)? None of the given choices

A: The solution for the above question is as shown below.

Q: Find the length of the curve x(t) = e" cos(t), y(t) = e" sin(t) for 0 <t < 3. Give an exact answer,…

A: Given curve: x(t) = e^(5t) cos(t) y(t)= e^(5t) sin(t)

Q: Find the length of the following two-dimensional curve. r(t) = (2 cos t,2 sin t), for 0stsT

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Q: Find the length of the curve. r(t) = (9t, 3 cos(t), 3 sin(t)), -5 sts 5

A: To determine the length of the curve.

Q: The length of the curve defined parametrically by: x = e'cost, y = eʻsint,0 <t <, is equal to:…

A: given parametric curve x=etcost     y=etsint

Q: The arc length of the the curve x =V9 - y2 - 3<y<3 is equal to Select one: О Зл O 9T О бл 18T

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Q: Find the length of the curve. r(t) = cos(3t) i + sin(3t)j + 3In(cos(t)) k, 0 stsn/4 Submit Answer

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Q: the of the curve d:laaR? Find detined by is čonstant. 870 Curiature a(t) =|1 Cost, 1 sint, o where…

A: A vector is a quantity that has both magnitude and direction. In general vectors are represented as…

Q: Find the length of the curve x(t) ešt cos(t), y = et sin(t) for 0 <t < 10. Give an exact answer,…

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Q: In which direction is the curve x = -2 sin t, y = 2 cos t, for 0 ≤ t &lt; 2π, generated?

A: x = -2 sin t, y = 2 cos t Table for values of x and y: t x y 0 0 2 π 0 -2 π2 2 0 2π 0 2

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

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Q: for the curre y=: f(n) = n*, 0 sM 32 find the curve'slength given by: 1. 2 It dn

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Q: Let Sketch the curve and find the length of this curve. r(t) = (t, 3 cost, 3 sin t), 0≤ t ≤ 2π.

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Q: let rit)= demti+ dsintf+? Find the length of this curve from t=0 to t = 4.

A: we have to find the length of the curve

Q: Find the curvature k of the curve r(t)=ti+ 6t^2j+4tk

A: To find: The curvature k of the curve rt=ti+6t2j+4tk. Formula used: The curvature of curve is…

Q: Find the length of the following two-dimensional curve. r(t) (2 cos t+2t sin t,2 sin t- 2t cos t),…

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Q: Find the length of the following three-dimensional curve. r(t)= (sin t, cos t, 2t), for 0sts4

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Q: The arc length of the curve x(t)=2sint, y(t)=2cost, Ost <2t is Select one: O 4T O None O 2T

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Q: What does the arc length formula give for the lengthof the line r(t) = ⟨2t, t, -2t⟩, for 0 ≤ t ≤ 3?

A: Given:          r(t)=⟨2t,t,−2t⟩  for 0≤t≤3.

Q: Find the length of the following two-dimensional curve. r(t) = (5 cos t, 5 sin ), for 0sts

A: Recall: Length of the Parametric form :L=∫t=at=bx'(t)2+y'(t)2 dt

Q: Find the curva tuie For He Curve 2+); +(6 cos 2t) + 5¢ K %3D

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Q: Find the length of the following​ three-dimensional curve. r(t)=cos t, sin t, 3t​, for 0≤t≤2

A: Given that rt =cost,sint,3t  for 0≤t≤2 find the length

Q: Find the length of the curve. r(t) = (3t, 3 cos(t), 3 sin(t)), -3 sts 3

A: For the curve r(t) = f(t) , g(t), h(t)On the interval [a, b]The length is given by L =…

Q: Determine the length of the curve r(t) = 4sin(3t)i + 3tj + 4cos(3t)k, 0 <t < -. 2 Give your answer…

A: Arclegth of the position vector

Q: Find the length of the curve. x- 5 sin t+ 5t, y= 5 cos t, 0 st< O A. 5T O B. 20 O C. 10 O D. 30

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Q: Find the length of the curve r(t) = (cos(3t), sin(3t), 4t) for -6 ≤ t ≤ 5 Give your answer to two…

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Q: Find the length of the curve #(t) = et cos(t), y = e* sin(t) for 0 <t < 5. %3D %3D Give an exact…

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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: