Complete each of the following. (a) Compute the arc length of the portion of the curve r(t) = (2t, t², In t) between the points (2, 1, 0) and (4, 4, In 2).

Q: Determine the length of the curve r(1) = t*i+ t°j – 5k, 4 <1 < 1. %3D Give your answer correct to…

A: Given curve is  r(t)=t4i+t6j-5k,    1≤t≤4

Q: Find the arc length (s) of the curve r(t) = (- e*, 2e, – 4e*) for 0 <t < In(3) S =

A: To find the arc length of the curve

Q: Find the arc length of the curve c(t) (3t, 18t) for -1 ≤ t ≤ 1. (Express your answer as an exact…

A:

Q: 8. Consider the curve 7(t) = (t², t³, 1) Find the arc length from t = 0 to t = 2.

A: r→(t)=t2,t3,1when t=0,t=2Find : Arc length

Q: Find the arclength of the curve r(t) = (6/2t, e¤, e -ot), 0 < t<1

A: Given  curve   rt=62 t, e6t, e-6t  0≤t≤1To determine  the  arc  length  of the curve  rt    rt=xt…

Q: Find the arc length of the curve c(t) (2t, 19t|) for -1 <t < 1. (Express your answer as an exact…

A:

Q: Calculate F.dr, where C is the curve given by ti + 123+ tk, with t = [0, 1] and F(x, y, z) = 4xei +…

A:

Q: Find the arc length (s) of the curve r(t) = (-41³/2 +3, -4t³/2+2, -4t - 4) for 2 ≤ t ≤ 4 S =

A:

Q: Compute the arclength of the following curves. (y-1)³ =x², for 0 <x< 5 а.

A: We have to compute the arc length for the function y-13=x2, for 0≤x≤5.

Q: Find the arclength of the curve r(t) = (10/2t, el0, e-10r), 0 <t < 1

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Q: (- 4t + 3, t + 2, – 2t – 4) for -4 <t < - 1

A:

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: 3 5 х3; [1,3] 3 1 (с) y = у 10 У — X3 2

A: Note:- Since you have asked multiple question As per our policy, we will solve the first question…

Q: Find f (x-y3)d a + xy dy if the contour C looks as follows: It consists of two arcs of radius 1 and…

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Q: Find the arc length (s) of the curve 7 (t) = ( – 2t + 3, 4t%/2 + 2, – 4t/2 + 6) for 1 <t< 4 S =

A: Arc length is-  Integration with limits a to b [√(dx/dt)^2 +(dy/dt)^2 +(dz/dt)^2] dt

Q: Find an arc length parametrization r₁ (s) of the curve r(t) = (71,3/2, 103/2), with the parameter s…

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Q: Find the arc length (s) of the curve 7(t) = ( - 2t + 2, – 2t3/2 + 1, – 4t3/2 4) for 0 <t < 3 - S =

A:

Q: Find the length of the curve y = V- from z =1 to r = e. 3

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Q: Find the length of the curve. r(t) = (7t, 3 cos t, 3 sin t), -2 ≤ t ≤ 2

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Q: Find the length of the curve. The parabolic 8 ,0 <0< 2 A) 4(v/2 – In(v2 – 1)) segment r = 1+ sin(0)'…

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Q: Consider the curve r = (e-¹t cos(-6t), e-¹ sin(-6t), e-¹t). Compute the arclength function s(t):…

A: Consider the curver = (e-t cos(-6t), e-t sin(-6t), e-t )We have to Compute the arclength function…

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.