Find the distance traveled by a particle with position (х, у); Cos t, y = (cos t)², 0 < t < 4TT X =

Q: Find the point on the curve r(t) (12 sin t)i – (12 cos t)j + 9tk at a distance a) 5n units and b)…

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Q: The arc length of the curve r= cos 30 with 0<0<nis попе J V-9(sin 30)² + (cos 30)² d0 I (sin 30)² +…

A: In this question we have to find the arc length of the curve.

Q: Find the length L of the curve R(t) = e* cos(3t) i + e* sin(3t) j + 3e*k over the interval [1, 5]. L…

A: given: Rt=etcos3ti+etsin3tj+3etk interval 1,5

Q: Find the length of the curve r = cos(f) + sin(0), 0sOs

A: The objective is to find the length of the curve.

Q: d the arc length of the curve y=(x^2)/2 - (lnx)/4 O

A: Explanation and solution is given below....

Q: y = sin(x) at |0, 0, with four equally – spaced points

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Q: A cantilever beam of length L is embedded at its left end. Find the deflection of the beam when w(x)…

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Q: Find the length of the curve. r(t) = cos(2t) i + sin(2t) j + 2 ln cos t k,     0 ≤ t ≤ π/4

A: Consider the length of the curve is, r(t)=costi+sintj+(lncost)k Then,…

Q: Find the length of the curve over the given interval. r = 1 + sin θ, [0, 2π]

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Q: Find the position function of the particle moving along a line that satisfies the a(t)=4 sint-cos t,…

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Q: Find the length of the curve over the given interval. r = 4 sin θ, [0, π]

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Q: Find the surface area of the figure generated by revolving x = 6 cos(t), y = 6 sin(t) where 0 <ts…

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Q: The value of cosh(3z+1)dz, where C is the line segment from z=1 C to z= 2i. O a. (-sinh(6i+1)+sinh…

A: We have to find the value of integral,  I = ∮Ccosh(3z+1)dz  ;  where C is the line segment from z=1…

Q: Find the length L of the curve R(t) = e" cos(2t) i + e* sin(2t) j+2e"k over the interval [1, 4]. L =

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Q: (x,() + x(x, t) = sin(t) u(n, 0) = cos(7), at %3D %3D nER.

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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: Determine the speed (in meters per second) of a particle with a trajectory (3 sin (3t), 8 cos (3t)),…

A: We have to find dsdtt0=π4

Q: A) Find the distance traveled by a particle with position (x, y) as t varies in the given time…

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Q: Find the point on the curve r(t) = (12 sin t)i - (12 cos t)j + 5t k at a distance 13pai units along…

A: Given,

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

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Q: Find the length of one arch of the cycloid 1 8(t- sin(t)), y = 8(1 – cos(t)).

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Q: Sketch the motion of a particle with position as: x = 5 sint y = ²/-nstsn - St≤n as tvaries in the…

A: We need to plot the motion of the particle, whose position is defined as: x=5sint, y=t2 in the…

Q: Find the length of one arch of the cycloid = 5(t-sin(t)),y- 5(1-cos(t)).

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Q: cos(t)y'-tsin(t) = y ; is this seperable or non-seperable?

A: It can be shows that it is not separable. See the next step

Q: Assume that z and y are both differentiable functions of t and are related by the equation = cos…

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Q: Please solve

A: Parametrization of tangent lines is given by

Q: Find the exact length of the curve. x = 9 cos(t) - cos(9t), y = 9 sin(t) - sin(9r), Ostsn 9 cos (47)…

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Q: Find the distance traveled by a particle with position (x, y) as t varies in the given time…

A: I am going to solve the given problem by finding the arc length of the given parametric curve to get…

Q: Find the length of the arc from 0 = 0 to 0 = 2n for the cardioid r = f (0) = 2 – 2 cos 0.

A: Solution if r=fθ  has a continuous first derivativefor α≤θ≤β then length of polar…

Q: 6) Find the equation of the osculating plane of the helix x = sin 3t, y = 2t, at the point (-1,л,0).…

A: Given :x=sin3ty=2tz=cos3tpoint -1, π, 0

Q: Find the exact length of the curve defined by x = 3 cos t – cos 3t and y = 3 sin t – sin 3t on the…

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Q: Compute cos(x) cos(y) dx – sin(x) sin(y) dy, where c(t) = (9t, t2), 0 sts 1. (Round your answer to…

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Q: Find the distance traveled by a particle with position (x, y) as t varies in the given time…

A: Given : x = cos2ty = cos t0 ≤ t ≤ 4π

Q: x = sin´t, y = cosʻt, 0<t<3™

A: Given x=sin2t, y=cos2t, 0≤t≤3π

Q: Find the length of the curve x = 8 cos t + 8t sint,y= 8 sin t – 8t cos t ; 0 < t %3D -

A: The length of the parametric curve is given by  L=∫abx't2+y't2dt.

Q: What is the integrating foc tor of the equation %3B

A: Topic:- differential equations

Q: Find the point on the curve F(t) - (5 sin t)i + (5 con t)j+ 12tk at a distance 20 units along the…

A: Let's find.

Q: Find the point on the curve r(t) = (5 sin t)i + (5 cos t)j+ 12tk at a distance 26™ units along the…

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Q: Find the distance traveled by a particle with position (x, y) as tvaries in the given time interval.…

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Q: Find the distance traveled by a particle with position (x, y) as t varies in the given time…

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Q: Find the arc length of the curves x = e-20 (sine + cos0), y = e-20 (sine – cose) over the interval…

A: Definition used -                    Length of the curve…

Q: Find the distance traveled by a particle with position (x, y) as t varies in the given time…

A: The objective of the problem is to find the distance traveled by the particle whose position is…

Q: A rod with density p(x)=4+sin(x) (in grams per cm) lies on the x-axis between x=0 and x=pi/3. Find…

A: Please see the explanation below

Q: The arc length of the curve r= cos 30 with 0<0<T is J V (sin 30)² + (cos 30)² d0 T" V-9(sin 30)² +…

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Q: Find the distance traveled by a particle with position (x, y) as t varies in the given time…

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Q: Find the intergral of  (t^2 sin(t)cos(t) +t sin^2 (t) )dt

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Q: 3) Find the exact length of the curve. y = In(cos r), 0 < x<

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Q: Find the length of one arch of the cycloid x= 6(t-sin(t)) y= 6(1-cos(t))

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Q: The arc length of the curve r= cos 30 with 0<0 ST is I V-9(sin 30)² + (cos 30)² d0 0. I V(sin 30)² +…

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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: