Consider the path r(t) (14t, 7t, 7 In t) defined for t > 0. Find the length of the curve between the points (14,7,0) and (56, 112,7 In(4)).

Q: ) Find the orthogonal trajectory of the family of curve given by y = In(r³ + b)

A: First formulate a differential equation to eliminate the parameter b . Then replace (dy/dx) by…

Q: Evaluate I = fe(sin z + 9y) dz + (7z + y) dy for the nonclosed path ABCD in the figure. А - (0,0),…

A: Given integral is I=∫Csinx+9ydx+7x+ydy For non-closed path, integration will be…

Q: Sketch the plane curve r(t) = 2ti − 3tj,  and find its length over the given interval [0, 5]

A: Given plane curve: The parametric form of vector valued function is It is the equation of line in…

Q: Prove that for any point (a, b) ≠ (0, 0) on the parabola y = x2 , the normal line intersects the…

A: The given equation of parabola is y=x2  For the point (a,b)≠ (0, 0), the slope of normal is and at…

Q: Let C C R? be the path from (3, –2) to (3, 2) along the graph of the curve x – 5x?y? – 7x2 = 0.…

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Q: 5. Prove that whole length of the curve x3 + y2/3 = a/3 is 6a.

A: Given that x23+y23=a23 To find the length of the curve:…

Q: has its vertex at the origin and has directrix at z = 38

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Q: Sketch the plane curve r(t) = 3ti − tj and find its length over the given interval [0, 3] .

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Q: Consider the path r(t) = (4t,2t 2 ,2lnt) defined for t > 0. Find the length of the curve between the…

A: Given the path r(t) = (4t, 2t2, 2ln t), t>0. Find the length of the curve (4, 2, 0) and (20, 50,…

Q: Let C, be the straight line path z = (1 + i)t, 0 Sts 1; Cz the quarter circle z =1- cost +isin t, 0…

A: Given that C1 be the straight line path z=(1+i)t, 0≤t≤1; C2 the quarter circle z=1-cos t +i sin t,…

Q: Evaluate I = Se(sin z + 4y) dz + (5x + y) dy for the nonclosed path ABCD in the figure. %3! A =…

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Q: Do the curves ri(t) = (t² + 3, t+1, t² – 5) and rž(s) = (4s, 2s – 2, s + 1) intersect? If so, at whi…

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Q: Sketch the plane curve r(t) = t2i − 4tj and find its length over the given interval [0, 5] .

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Q: Q4) Compute f z'dz, where C is the path formed by the line segment from -3 to 3 and the semicircle…

A: There is no such introduction for the solution.I have explained it within the solution.Please go…

Q: 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→To calculate:- ∫C2F→.dr→where C2 is the line segment starting from…

Q: Find the area of the inner loop of the curve r= 1-2cos θ

A: For the inner loop of the curve 1-2cos θ=0⇒2cos θ=1⇒cos θ=12⇒θ=cos-112⇒θ=π3,5π3

Q: Suppose that these two lines are the paths of two particles. If t and s measure ime in seconds, and…

A: We have given the equations of two lines : l1 : x,y,z=2+2t,8+t,10+3t l2 : x,y,z=5+s,12-2s,17-s…

Q: Determine the distance between: the pa rallel line x-a++ X - 3+S+t t9= 5-t,tER and the…

A: Given line is                                                     x=2+ty=5-tz=-4 point on the line…

Q: Sketch the plane curve r(t) = t3i + t2j and find its length over the given interval [0, 1] .

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Q: Find the exact length af te of the curve, 2. x-

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Q: Consider the curve r (t) = (t, tº, tº) ; t > 0 Does the point (1, 1, 1) belong to the curve and if…

A: Please refer the attached image for complete solution.

Q: 2) Consider the oriented curve C given by C = C1uC2, where C2 is obtained by intersecting the…

A: We will first parametrize the curves C1 & C2 then substitute the x,y,z values into the line…

Q: Consider the inmplicitly defined curve xy +3x2 4. %3D O Verify that the point (1, 1) is on the…

A: Consider, the implicitely defined curve                          xy2+3x2=4 We have to verify the…

Q: Prove that the curve y: (-,-)→ R³ is regular with factor H and determine whether the image y (t) =…

A: If γ't not equal to 0 for all t∈-π2,π2then it is regular. If γ''t=kγ't for all t∈-π2,π2 then the…

Q: Compute the length of the curve r(t)= (-3t, 3t + 7,–7t + 1) over the interval 0 <t <7

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Q: By using Green's theorem evaluate (4.ry – y') dr + (x + 3ry) dy where C is a closed curve starting…

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Q: where Projuct of in tegra tion SÃr dR A-zi-xj-YK and =dxi +arj +dzK dR and %3D the seguent of the…

A: Use line integral

Q: compute the length of the curve over the given interval. r(t) = 2ti − 3tk, 11 ≤ t ≤ 15

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Q: Sketch the plane curve r(t) = ti + t2j and find its length over the given interval [0, 4] .

A: Concept: The calculus helps in understanding the changes between values that are related by a…

Q: find the arclength of the path (P- 1.-2) for 0<t< 4

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Q: Compute the length of the curve r(t)= (-5t, 4t – 3, – 5t + 5) over the interval 0 <t< 7

A: Disclaimer: Since you have asked multiple questions, we will solve the first question for you. If…

Q: Given that the line y =r- b is an asymptote to the ar curve y a,b €R,r = 1, then b = I-1 -1-a B 1-a…

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Q: Evaluate ∫C x ds, where C is the parabolic curve x = t, y = t2, from (0, 0) to (2, 4).

A: Given: Parabolic curve x=t, y=t2 from (0,0) to (2,4)

Q: Let C be the curve r(t) = (2t, t², t°/3). (a) Compute the Frenet frame for C. (b) Calculate the…

A: The position vector is rt=2t,  t2, t33. Differentiate with t and obtain the unit tangent vector:…

Q: Evaluate I = Se(sinz + 2y) da + (8r + y) dy for the nonclosed path ABCD in the figure. %3D А 3 (0,…

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Q: Let C be the part of the curve oriented counter- clockwiso Evaluate xtyal with Fidr,

A: Here we will use stokes theorem to find given line integral for vector F  ∫CF.dr=∫∫SCurlF.ds S is a…

Q: Show that the curve a(t) = (t, , ) lies in a plane and find the equation of this plane.

A: For all 't' , we have α'(t) = 1 , -1t2 ,-1 - 1t2α"(t) = 0 , 2t3 , 2t3 Now  α'(t) × α"(t) = 2t3 ,…

Q: Let C be the curve connecting (0,0,

A: Given that C be the curve connecting  (0, 0, 0) to (1, 4, 1) to (3, 6, 2) to (2, 2, 1) to (0, 0, 0).…

Q: Let C2 be the curve defined by R(t) = (t² +t + 3, 4t – 5, 2+ t – t2), t e [0, 1]. %3D | (x + y+ 2)…

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Q: Consider the implicitly defined curve xy? +3x2 = 4. %3D Verify that the point (1, 1) is on the…

A: Consider the following curve: xy2+3x2=4 Substitute x=1 and y=1 in the above curve:…

Q: Di+Oj

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Q: Find the length of the following​ three-dimensional curve.   r(t)=2+3t,6−2t,−8+7t​, for 1≤t≤6

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Q: Find the exact length of the loop of the curve = 3t – t°, y = 3t2 -

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Q: Draw the isoclines with their direction markers and sketch several solution curves, including the…

A: According to the given information, it is required to draw the isoclines with direction makers and…

Q: Consider the intersection curve C in the first octant, between the surfaces s1: 2 = 1- r? S2 : 2 +…

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Q: Consider the path r(t) = (4t, 2t², 2 In t) defined for t > 0. Find the length of the curve between…

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Q: Sketch the plane curve r(t) = t2i + 2tk, and find its length over the given interval [0, 3]

A: We have, r(t)=t2i+2tk  we have to determine the length of the curve over the interval [0,3]

Q: Let C2 be the curve defined by R(t) = (t² + t + 3, 4t – 5,2 +t – t²), t e [0, 1]. Evaluate | (x + y+…

A: The solution is given as

Q: Let C1 be the straight line path z = (1 + i)t, 0 < t S 1; Ca the quarter circle z = 1- cost +i sin…

A: