2. Find the length of the curve 1

Q: (1) Evaluate the line integral 4xy dx + (2x² – 3xy)dy if the curve C consists of the line segment…

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Q: (a) Find the arc length of the curve y = (4 – x3 over the interval [0, 8].

A: " Since you have asked multiple question, we will solve first question for you. If you want any…

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: 2. Eliminate the parameter and find the equation of the curve in x,y Curve: x=1-sin?t, y= cost, 0 <t…

A: topic - equation of curve in x,y

Q: 2:(b) Find the exact arc length of the curve x=e"(sint+ cost), y=e" (sint – cost) over the interval…

A: Given

Q: 3. Consider the parameteric equations for a curve C given by x(t) = cos t(1 – cost), y(t) = sin t(1…

A: Given that: xt=cost1−costyt=sint1−cost

Q: 3 The arc length of the portion of the curve y =x² with 0<x<4 is

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Q: Find the arc length of the graph of the function over the given interval:

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Q: Find the length of each of the following curves: %3D %3D x (ey + e) from y = 0 to y = 1

A: The length of a curve means the arc of the length of a curve. The arc of the length of a curve in…

Q: Find the length of the arc of the curve y =: +e*) from x = 0 to x =2.

A: y=ex+e-x2

Q: 10. For the asteroid given by = 4cos (t), y = 4sin' (t) on 0,2n|, find the slope of the tangent at…

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Q: - Sva cos 2t dt from x= 0 to x= Find the length of the curve y =

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Q: 1. Find the moment of inertia, with respect to the line x=1, of the area enclosed by the loop of the…

A: Given: Curve y2=x2(1-x)----(i)

Q: 1. Evaluate xyz dz, where C is the curve 7(t) =t cos(t)î + sin(t)ĵ – cos(t)k, 0 < t <1.

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Q: 4. Find the length of the arc of the curve 6xy=yt +3 from the point where y - 1 to the point…

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Q: 1) Find the arc length of the curve over the interval: | 2 3 (a) [0,3] for f (x) =x X2 (b) for y =…

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Q: 10. Find the exact length of the curve y = In(sec x),0 < x<T/4 %3D

A: This is a problem related to integration of the function. Based on the integral formula we will find…

Q: compute the length of the curve over the given interval.  r(t) = (cost,sin t,t3/2), 0 ≤ t ≤ 2π

A: Derivative of a function is represented as the change in variable with respect to change in another…

Q: 3 Consider cune detined by y cos (T2) =UY +y+2 2 ) Verify point (2,-2) belongs to this curve, fond…

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Q: For the curve given by R(t) = (cost + tsin t)i + (sin t − t cost)j + (t^2)k, find a. the length of…

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Q: а. z = t', y = ; 0 < t < 1 b. x = cos (2t), y = sin°(2t); 0 <t< 4

A: In the given question we have to find the exact length of the given curves.

Q: 4. Find the radius and centre of curvature of the curve x = (1 - at) cos t + a sin t, y = (1 – at)…

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Q: 1. Find the arc length of the curve x = r=(y²+ 2)2 from y-0 to y-1

A: Since you have asked multiple questions in a single request so we will be answering only first…

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: 2. Find the length of the curve r(t) = e'i + e' sin 2tj + e' cos 2tk, 0st< 2n.

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Q: 3 The arc length of the portion of the curve y=x 2 with 0<x<4 is

A: I have append the formula used to find arc length in the solution part.

Q: 2. Determine the arc length of the given curves a) y =+ 1<y< 2 3

A: s=∫y1y2 1+x'2 dyy'=3y23-14y2y'=y2-14y2s=∫121+y2-14y22dys=∫121+y4+116y4-2×y×14y dys=∫12y4+116y4+12…

Q: i) Find the length L of the curve y = x + cos x, 0<x <T/2 Use any one of the following formulas 2…

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Q: V. The tangent line at the point (t, y) = (- to 4'2 1 Then, the curve y = cos t is y = bt - - - a a…

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Q: Consider the following. x = sinh(t), y = cosh(t) (a) Eliminate the parameter to find a Cartesian…

A: Solution: The objective is to eliminate the parameter to find a Cartesian equation of the curve

Q: 1. Find the length of the curve r(t) = (sin 2t, cos 2t) from t 0 to t= 2n. CoS %3D

A: Since you have posted multiple question in this question which are not interlinked so i have solved…

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

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Q: 2 4. Find the length of arc of the curve x3 + y = 1 in the first quadrant from the point where x =…

A: Let's find arc length of given curve in first quadrant.

Q: A curve is written parametrically with x and y defined as functions of t: x=t2 +1, y = 4r° +3. Find…

A: Take the derivative of x and y with respect to t then use the formula for the arc length.

Q: 9) Find the exact (arc) length of the curve. x = 2t – sin(2t), y = 1 – cos(2t), 0<t<r. -

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Q: 4. Find the arc length of the curve x=y+y from y-1 to y-4

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Q: #4

A: Find the tangent plane to the given surface at the specified point

Q: 2. Find the arc length of the curve y = 16 from x=2 to x-3 2x

A: Given y=x416+12x2, x=2 to x=3

Q: - Find the aera under the curve y = cosx Form x = 0 to x = -fx sec² (ax² )dx a#0

A:

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: Find the length of the curve 7 (t) = (4 cos(t), 4 sin(t), 3t) for – 1 <t< 4

A: Here we have to calculate arc length in given interval.

Q: Determines the length of the curve in the indicated interval x = 3 sin t+ 3t, y = 3cos t, 0stsm O 12…

A: Given curve is x=3sint+3t and y=3cost, 0≤t≤π. We have to determine the length of the curve on the…

Q: 1) The Arc Length of the curve f(x) = coshx, 0≤x≤ In3 is: d. 312 12 3 C. b. 2²/2 0.33 lorod by…

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Q: Find the length of the curve. r(t) = cos(2t) i + sin(2t) j + 2 In(cos(t)) k, osts n/4

A: Given:

Q: 1. Find the arc length of the curve f(x) = 3 4 (x + 2)2 on the interval [0, 4]. 3

A: Find the arc length

Q: For the smooth curve, find any points where the tangent line is either horizontal or vertical. x(t)…

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Q: et +e- find the length of the curve h(x) = from r = 0 to z = In 2.

A: The standard formula for arclength will be used in solving this problem. The typeset solution is…

Q: 3) Find the arclength of the following curve on the given interval. x = 31, y= 41; 1<t<4 X =

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Q: Find (2x + xy)ds, where C is the arc of the curve y = sin (x) for 0 ≤ x ≤ π

A: The given problem is to evaluate the given line integral of the given function over the given curves…

Optimization

Optimization comes from the same root as "optimal". "Optimal" means the highest. When you do the optimization process, that is when you are "making it best" to maximize everything and to achieve optimal results, a set of parameters is the base for the selection of the best element for a given system.

Integration

Integration means to sum the things. In mathematics, it is the branch of Calculus which is used to find the area under the curve. The operation subtraction is the inverse of addition, division is the inverse of multiplication. In the same way, integration and differentiation are inverse operators. Differential equations give a relation between a function and its derivative.

Application of Integration

In mathematics, the process of integration is used to compute complex area related problems. With the application of integration, solving area related problems, whether they are a curve, or a curve between lines, can be done easily.

Volume

In mathematics, we describe the term volume as a quantity that can express the total space that an object occupies at any point in time. Usually, volumes can only be calculated for 3-dimensional objects. By 3-dimensional or 3D objects, we mean objects that have length, breadth, and height (or depth).

Area

Area refers to the amount of space a figure encloses and the number of square units that cover a shape. It is two-dimensional and is measured in square units.