1 Let C be the curve y = x° + for 0.2 < a < 5. 12x 120 108 96 84 72 60 48 36 24 12 05 i 15 2 2,5 3 3,5 4 4,5 -12 Find the arc length of C shown above. First find and simplify /1+ y² Then arc length =

Q: Given x = y² +y + 1 from 0≤y=2 The Arc Length of the given curve over the interval is a. 7.38 units…

A: I have stated the used formula in the solution

Q: 12. Evaluate 3a ds where C is the parabolic arc along y = from P(0,0) to Q(2, 2).

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Q: a) Find the arc length parametrization of the line x = t, y = t that has the same orientation as the…

A: a)The given line is x = t, y = t.We can write it as a vector valued function as r(t)→ =x(t) i^ +…

Q: Let C be the curve y for 0.3 <z< 4.9. = 4 27 24 21 18 15 12 0.5 1 1.5 2 2,5 3 3.5 4 4.5 Find the arc…

A: SOLUTION-

Q: Given r(t) = et sin t i + et cos t j + v2etk, a) find the arc length function for the curve measured…

A: (a) Given a curve rt=etsinti^+etcostj^+2et k^, to find the arc length of the curve measured from the…

Q: 2 3/2 1 1/2 on [1,9]. Find the arc length of the curve y :

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Q: Let C be the curve y = 4 In(16 – r), for –1 <r < 0.5. A graph of y follows. 11.1- 11.07 11.04 11.01…

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Q: Consider F and C below. F(x, y) = x2 i + y² j C is the arc of the parabola y = 2x2 from (-1, 2) to…

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Q: 1 for 2x2 0.1 < x < 3. The graph follows. Let C be the curve y 16 ●50 45 40 35 30 25 20 15 10…

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Q: The arc length of the curve x = 4(3 + y)² on [1, 4] is approximately (i) (ii) (iii) None of these…

A: Explanation of the solution is given below...

Q: Find the arc length of the graph of y = (x3/6) + (1/2)x on the interval [ 1/2 , 2], as shown

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Q: 3. Let F(x, y) = x²e* i + 60x²y² j, and let C1 denote the curve consisting of the line segment from…

A: Given: F→(x,y)=x2exi→+60x2y2j→

Q: Find the arclength of the curve r(t) = (2 sin t, 6t, 2 cos t), –7 < t < 7

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Q: Find the arc length of the curve below on the given interval y=1/3(x^2+2)^3/2 on [3,6]

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Q: 1. Evaluate dz and dz where C is a clockwise circular arc that begins at – 2i and ends at 2i. Show a…

A: We have to integrate the given complex integrals.

Q: Find the exact length of the curve x = t sin t y = t cos t for 0 <t< 1. You may use the following…

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Q: Let C be the curve y for 4г10 120 0.9 < x < 2.4. The graph follows. з00 270 240 - 210 180 150 120 90…

A: Arc length of the curve is given by:

Q: Find the length of the curve r(t) = (3 cos t)i + (3 sin t)j + 2t³/²k , 0sts3. %3D

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Q: Let C be the curve y 3.5 In (12.25 "), for- 3.4 < < 0.2. A graph of y follows. 7. 4 3. 12. -3. -2.5…

A: Given: y=3.5ln12.25-x2, for  -3.4≤x≤0.2 We have to find length of the arc. The sum and difference…

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: Consider the space curve ř(t) = (8 sin(5t), 2/7 cos(5t), 6 cos(5t)). %3D a. Find the arc length…

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Q: Find the exact arc length of the curve x = y4/8 + 1/4y2 ,   1 ≤ y ≤ 3

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Q: Consider a curve given by Ř(t) = (2 cos t, 12t + 3, -2 sin t). Let f(t) be a function such that f(1)…

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Q: Consider the space curve r(t) = (5t + 2, – 2t + 2, 3t + 2). | a. Find the arc length function for…

A: The given space curve is: r→(t)=5t+2, -2t+2, 3t+2

Q: that the reparametrizatIon by arc-length traverses the curve with unit speed. That is, snow ||7…

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Q: Let C be the curve y 3.5 In(12.25 r"), for-3.4 < K 0,2. A graph of y follows. 0.2 Find the arc…

A: Arc length

Q: The arc length function for the curve r(t) = (3e' , 2, 4e*), 0 <t< 1, is s(t) =

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Q: Let C be the curve represented by R(t) = (cos 2t, 2√3t, 5 — sin 2t). Find the coordinates of point Q…

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Q: Let C be the curve y for 0.7 SaS 1.8. 16 1.5 14 12 0.8 06 05 04 Find the arc length of C shown…

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Q: V. Let C' be the curve represented by Ŕ(t) = (cos 2t, 2√√3t, 5 — sin 2t). ㅠ 1. Determine the…

A: Hint:  The formula to obtain the curvature is κ=∥T'(t)∥∥r'(t)∥, where T(t)=r'(t)∥r'(t)∥.

Q: Let C be the curve y = 12 11 10 9 8 7 6 5 4 15 x³ 1 + 4 3x 25 Find the arc length of C shown above.…

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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: Prove that e dz < ++1 3 where C is the arc of the circle |2| = 3 from z = 3i to z = -3.

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Q: Find the length of the arc of the curve x = t, y= t° between the points for which t = 0 and t = 2.…

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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: 3. Let C be the part of the space curve represented by r(t) = (t, t/², t² +1), where 1 <t< a. Find…

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Q: Let C be the curve y for 0.5 < a < 2.7. The graph follows. 2x2 16 3.3 2.7 2.4 2.1 1.8 Y.5 0.9 0.6…

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Q: Find the length of the following two-dimensional curve. r(t) = (12t – 2,16t +2), for 0sts1 The arc…

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Q: Find the arc length of the parametric curve. x = e', y= et, z = /2t; 0<t < 8 ||r'(t) x r"(t)|| Use…

A: The arc length formula for a parametric curve is L=∫abdxdt2+dydt2+dzdt2dt. The power rule of…

Q: Determine the arc length of the function f(x) = coshx on the interval [0, 2]. A None of the given…

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Q: Find the length of the following two-dimensional curve. 兀 r(t) = (9 cos t + 9t sin t 9 sin t-9t cos…

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Q: 0) have the form: a + bt %3D y c+ dt %3D etric curve starts at (7, –6) when t = 0 and ends at (1,-6)…

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Q: Find an arc length parametrization of the curve r(t) = (1/1+t^2,-1,t/1+t^2)

A: Find an arc length parametrization of the curve r(t) = (1/1+t^2,-1,t/1+t^2)

Q: Find the arc length of the curve y = e® from x=0 to x=1. VIte-1 VIte+1 - (v2 + ¿ln a. V2+1 Ob.…

A: Consider the equation: y=ex……1 from the limit x=0 to x=1.

Q: 3. Find the arc length of the curve x = (1+t)², y = (1+t)*, 0 < t < 1. 4. Find the arc length of the…

A: 3.

Q: Let C be the curve y = 4.5 In(20.25 – a"), for -3.6 < a < 4.1. A graph of y follows. 14 13- 12 10…

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Q: Find the arc length of the graph from A to B. f(r) = r/2 +1 24 22 20 18 16+ 14+ 12+ 10 8. (4, 9) 6.…

A: the arc length L=integration A to B of √(1+f'(x)^2)dx.

Q: 1. Given x = y² + yt! X y²+ y +ı from osy≤2 Which of the following expression gives the arc length…

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Q: 5 for 0.2 < x < 2.3. 12x Let C be the curve y 5 2.6 2.4 2.2 1.8 14 0.8 0.6 0.4 0.2- 0,5 1.5 Find the…

A: The arc length of a curve is given by- L=∫ab1+y'2dx The given function is- y=x35+512x where…

Q: Find the arc length function s(r) for the curve y² = r³, y 2 0 starting at the point (1, 1) and use…

A: Consider the curve  y2=x3 Starting point (1,1)

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.