Show that 22 dz = 0 1 – cos(2) whenever y is a closed simple smooth curve contained in a very small disc around the origin but not passing through the origin.

Q: Find the exact value of 0.049768 x³y²z ds, where C is the curve with parametric equations x = e¯t…

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Q: Find the arclength of the curve z 9 cos(9t), y = 9 sin(9t) with 0 <t< 36

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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: Show that the curves r = a sin 0 and r = a cos 0 intersect at right angles.

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Q: Find the points on the cardioid r = 1−cosθ at which there is a horizontal tangent line, a vertical…

A: Consider the given: r=1−cosθWrite the cartesian…

Q: Show that the curve y = x? sin 3x touches the parabolas y = x andy =-x² alternately at infinite…

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Q: ne if the curve (In t. t In t. 1) is tangent to the surface rz- uz? + cos Tu =1 at (0.0. 1).

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Q: Let C be the circle of radius 3 given by r(t) = 3 cos ti + 3 sin tj, 0 st s 2n as shown in Figure…

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Q: 2. Write an arc length parameterization for the curve r(t) = cos ti + sin tj + 3t k ; 0<t

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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: Evaluate dy/dx at t = 1 for the parametrically defined curve with X = t3 and y = 5 - t. O -1.5 O 1.5…

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Q: Consider the curve C shown in the figure, parameterized by: C: r(t) = (1 - cos(t), 1- sin(t), 1 +…

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Q: Show that the exact length of the curve below is y = Vr - r² – sin ¬1 (V7)

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Q: (a) (., Consider the curve defined by ry = In(3x + sin(y)). Use implicit differenti- ation to find…

A: Consider the given equation of the curve, xy=ln3x+siny Use the implicit differentiation,…

Q: Find the arclength of the curve x = 3 cos(5t), y = 3 sin(5t) with 0 <t< 20 %3D

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Q: Let C be the parametric curve defined by tan-1t y = } In(1+ t²) where t > 0. dy (a) Find and dx dx2…

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Q: Find the arclength of the curve y = * - In x if < y s-1. Do not simplify your answer for this…

A: Given: y=18x2−lnx18≤y≤e28−1

Q: Find Sc F · dr for F = 3yi – (sin y)j on the curve counterclockwise around the unit circle C…

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Q: 7. (~Ch16 Review #6) Compute y3 dx + x² dy where C is the arc of the parabola x = 1- y² from (0, –…

A: In this question we need to find the given line integral .

Q: Consider a new curve Č parametrized by r(t) = (cos t esint, sin t- cos t), 0 <t <I but with F still…

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Q: Consider the curves:r = 4sin 0 and r = 2 sin(0) + 2cos (0).

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Q: A particle moves in the xy-plane in such a way that its path is defined by x = e^t cos t and y = e^t…

A: Given that, x = e^t cos t y = e^t sin 2t Differentiating x with respect to t we get, dx/dt = e^t cos…

Q: Consider the parametric curve: x = ln (8t) y = 5t³ Which integral gives the arc length of the curve…

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Q: Find the arclength of the curve x 5 cos(3t), y = 5 sin(3t) with 0 <t< 6

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Q: Let 3 x = cos' t, y = sin³ t, z = 7; 0 < t < r. Find the arc length of the parametric curve.

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Q: 1. Find the moving trihedral and the curvature at any point of the curve R(t) = cos t î + sint ĵ+ k.

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Q: Find the arclength of the curve r = 5 cos(6t), y = 5 sin(6t) with 0 <t<. %3D

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Q: 9.40. Show that the 0-parameter curves on the surface x = (r cos 0)e¡ + (r sin 0)eg + f(0)eg are…

A: We have to show that the θ-parameter curves on the surface x are parallel.

Q: Find the points on the cardioid r = 4(1 – sin0) at which there is a horizontal tangent line, a…

A: According to the given information, it is required to determine the horizontal tangent line,…

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: (11) Find the tangent line(s) to the parametric curve given by * = t5 - 4t3 y = t2 at (0, 4)

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Q: Calculate the length of the curve r=a*sin(^3)(x/3) for (x(1)=0, x(2)=3pi)

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Q: Prove that all normal to the curve x =acid+at sin t, Y=a sin t-atcost.Are at a distance a from the…

A: We need to prove that normal to the given curve are at a distance of a from origin.

Q: Explain why a cubic Bézier curve is completely determined by x(0), x (0), х(1), and x'(1).

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Q: 4. Find an equation for the tangent plane to the surface z = 3x + 5y + arctan in the point (0, 1,…

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Q: Find the exact length of the curve. y = 2 In(cos (÷)) Tt/3 < x <T/2

A: Here given Differentiate the given function with respect to x using the chain rule of derivative

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

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Q: Compute the length of the curve where 0 ≤0 ≤ T. x = 2 cos² 0, y = 2 cos 0 sin 0,

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Q: Find the arclength of the curve z = 7 cos(3t), y = 7 sin(3t) with 0 < t< 12

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Q: 11. C is the curve F(t) = ( In (2 Vt +3 Find parametric equations for the tangent line to C at the…

A: Note: Here, the given point (0,1,2) is incorrect there should be(0,-1,2) because the point is on the…

Q: Calculate the length of the curve γ (t) = (cos(2t), sin(2t)) t ∈ [0, π].

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Q: Consider the parametric curve: X = sin² (t) y = -9t Which integral gives the arc length of the curve…

A: Topic = Integration

Q: 2. Let y be the curve in R3 given by t (t, 2 cos t, 2 sin t) for 0 <t < 47. Compute ds

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Q: Consider the curve y = – In|cos(x)| as x travels from T to 6 a) Find the length of the curve.

A: Hi, since you have asked multiple questions, we will solve the first question for you. If want any…

Q: Find the lengthof the curve r= a sin (0/2), 0s 0S T, a>0

A: The formula for finding arc length is L = ∫0πr2+ drdθ2dθ, where r = a sin2(θ2)

Q: Let C be the closed curve starting at (0, 1) traveling along the parabola y = (x – 1)? up to the…

A: Using Green's Theorem we solve the above question.

Q: The position of a particle in the xy-plane at time t is given by r(t) = (cos 2t) i+ (2 sin 2t) j…

A: Given: rt=cos2ti+2sin2tj For finding equation in x and y whose graph is the path of the particle, we…

Q: Consider the parametric curve: X = = 7et y = 3t5 Which integral gives the arc length of the curve…

A: I am going to solve the problem by using some simple calculus to get the required result of the…

Statements and Logic

Logic is the analysis of general patterns of thoughts without regard to any specific sense of context.