Which one is the formula for the arc-length function of a smooth curve r(t) for t>a. ||r" (u)|| du T'(u) du dT ds T'(t) |TO|| T'(t) O B(t) = T(t) × N(t)

Q: Calculus question in photo

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Q: 4. Find the arc length of function y = x2+ x- 2, -2 SXS1

A: Given: y=f(x)=x2+x-2  and -2≤x≤1 We have to find the arc length of the functionf(x).

Q: x2 The arc length function of the curve y = 2 - In x taking the point Po(1, ) CL as a starting point…

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Q: 22. Calculate the arc length of the graph of (a) y = f(x) = 1⁄2x³ + x-¹ over [1,3]. Hint: (b) y = 1…

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

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Q: 2. Find the arc length of the following curves on the given interval. x2 (a) y = 3 In x - 24 on [1,…

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Q: 5. Compute the arc length of the curve f (x) =1+In(cos(x)) over the interval from x=0 to X =

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Q: Consider a particle moving along a path C given by r(t) = v3ti+ = V3ti sin (t)j + cos (t)k Find the…

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Q: a) Show that the length of the arc of the curve y=f(x) where S(x) = Insecx 0SxS is 4 In(1+ v2)

A: We need to show that the length of the arc of the curve y=f(x) where f(x)=ln(sec(x)), 0≤x≤π4. The…

Q: Find the arc length over the given interval. 8. r(t) = (cos(3t), sin(3t)) [0, z] over 9. r(t) =(-t,…

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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: x? In x Find the arc length of the curve y = f(x) = 4 on the interval [1,2]. 2

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Q: 1) The arc length of the curve y = f(x) Over the interval asx<b is Length = +(s'x)}° de Approximate…

A: NOTE: According to guideline answer of first question can be given, for other please ask in a…

Q: Find the arc length of the curve f = vx from x = 0 to x = 4.

A: Consider the given function. f=x Find the derivative of the function. f'x=12x The arc length formula…

Q: Compute the trapezoidal approximation T5 to the arc length s of y = tan x over [0, 4].

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Q: 4. Find the exact arc length over the interval. b. x =y* from y = 1 to y = 4

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Q: X = 2t-2 - y = ln(t³) Which integral gives the arc length of the curve over the interval from t = 2…

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Q: x? In x Find the arc length of the curve y= f(x)= 4 on the interval [1,2]. 0.1

A: We will use integration to find arc length

Q: 8. Find the arc length of y3 + y = x² + 6x on the interval [-5, –1].

A: Given equation of the curve is y3+y=x2+6x                     (i) To find the arc length in the…

Q: 2. Find the arc length of (x+5 y= over the closed interval [-1,4] .

A: Given curve is

Q: - Find the arc length of f(x) = (-) (x² + 2)ž on the interval [0,9)

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Q: 4. Evaluate the line integral f x siny dx + y dy, where C is the arc of r(t) = ti + t²j from point…

A: Line integral

Q: Q: Prove that the arc lenfth af Curve y = log secx from to n- 3 is Use formula Note I+ ( dn.

A: Given: y = log sec x To prove: The arc length of the function y from x = 0 to is .

Q: 7. Find the arc length of the following plane curve: x = e' (sint + cos t), y = e' (cost – sin t),…

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

A: 1)- Given: r→t=sin 2t, cos 2t from t=0 to t=2π. The objective is to determine the length of the…

Q: Let f(x) = - 1. Calculate the arc length of the graph of r = g(y) over the interval z = 2 to a = 5.…

A: The given function is f(x)=x2-1 and the interval is x=2 to x=5 To find the arc length of the graph…

Q: 3 6. Find the arc length of f(x) = G (x2 + 2)z on the interval [0,9]

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Q: Express the arc length of the curve y =+ between x= -3 and x = -1 as an integral. A dx. E) None of…

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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: 5. Compute the arc length of the curve e f (x) =1+In(cos(x)) over the interval from x=0 to 4

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Q: The arc length of a curve y = g(x) over interval a s xsbis given as: S. VI + (g'(x))²dx Approximate…

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Q: Find the arc length of the curve below on the given interval by integrating with respect to x. x3 1…

A: Given the curve y=x33+14x ; x∈1,3

Q: Compute 2z dz (2² – i)3 when y is any arc z = z(t) in C with |z(t)| > 2 for every t, with…

A: Let the required arc be ABC, which starts at z=3 and ends at z=2i: The shaded area shows a sector…

Q: The arc length of the curve defined by the equations x(t) 10 cos(11t) and y(t) = 3t for 1 <t< 5 is…

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Q: 3. Given that r(t) = 3 sin ti+3 cos tj+4tk (1) Find the length of the curve for 0 <t<T/2 (2)…

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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: (a) Find the arc length function for the curve y= In (sin x) ,0 &lt; x &lt; pie  with starting point…

A: To find the arc length.

Q: Find the arc length of the curve y :In(cos x) from x = 0 to x = t/3. Length:

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Q: ind the arc length of the curve f(x) = x- between x=1 and r= 5. Note that 1+ (f'(1))- is a fect…

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Q: 4. Find the arc length of the curve y* +y from y31 to y-4 X%3D

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Q: For the function f(x)=1/4e^(−x)+e^x, prove that the arc length on any finite interval has the same…

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Q: Which one is the arc-length of the curve y =2/7x + 11, x E [7, 11]? 147x+280 dx 7x+ 11 14x2 +…

A: The curve y=27x+11 , x∈7,11

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: 3. a) Show that the length of the arc of the curve y=f(x) where f(x) = Insecx 0sxs is 4 In(1 + vz)…

A: L= integration of [1+{f'(x)}2] from a to b f(x) = ln sec x  f'(x) = tanx   Putting the value of…

Q: Which integral gives the arc length of the curve y = e2x on the interval 2 < x < 4? a)o (1+2e**)dx…

A: Given,  The curve  y = e2x on the interval 2≤x≤4  We have to find the integral which gives the arc…

Q: b) y = ln (cos x) 0 <x<- 3

A: Given: y=ln(cosx) in the interval 0≤x≤π3

Q: The integral V1 + ešx dx represents the arc length of which curve (for 0<x<1) ?

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Q: Q1. Find the length of the of the curve y = In(cos x) from x = 0 to x = . %3D

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

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.