) Find the gavea d the rgion enclased by the line x-In3 and the curves y= ex and y=ex

Q: Let C be the curve y = x³ – 3x2 + 4x - 1 joining the points (1, 1) and (2, 3). Find the value of |…

A:

Q: 3/2 Find the length of the curve. y = (x* +4)" OSxs2

A: see below the answer and calculation

Q: Let C be the closed curve indicated in the diagram. Use Green's Theorem (2,1) (o,i) (x² + y?, y³ +…

A:

Q: Let C be the curve y = x³ – 3x? + 4x – 1 joining the points (1, 1) and (2, 3). Find the value of…

A: Solving the Cauchy integral using Cauchy theorem

Q: 1. Consider the curve C given by y = 5 + 2x3/2, 0 < x <1. (a) Find the length of the curve C.

A:

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: Find the exact length of the curve. = 플Vy (y-3), 4sys 16 3

A:

Q: 2. Prove that the following curves are intersecting with a right angle: 5y-2x+ y –x*y =0 2y+5x+x*…

A: Given curves are  5y-2x+y2-x2y=0and2y+5x+x4-x3y2=0

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: 1. Find the are length of the curve x (+ 2)/2 from y=0 to y-1

A:

Q: Now compute [(x − 3y² + 2) ds where C' is the union of the curves C1 and C2. C1 is the line segment…

A:

Q: Find the exact length of the curve. x = et − 9t,    y = 12et/2,    0 ≤ t ≤ 2

A: Note that:Arc length formula = ∫abdxdt2+dxdt2 dt                   where , a≤t≤b       Given,…

Q: 2人+人 2X PM the From Pi (x,,y,z,) to Pz (xz, Yz, Zz Prove that the Midpoint of the line Ssegment 12…

A: Let the 2 points in the line segment be,

Q: 6a. Find . F dr, where F(x, y) = , and C consists of the circle x' + y = 4 from (2,0) to (v2, v2)…

A:

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 &amp; C2 then substitute the x,y,z values into the line…

Q: 7. Given that the line y = r -b is an asymptote to the ar curve y a,b €R,rz 1, then b = A -1-a 1-a C…

A:

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: 19. Evaluate Scx ds, where C is a. the straight-line segment x = t, y = t/2, from (0, 0) to (4, 2).…

A: Part (a)

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

A:

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

A:

Q: Let m be a positive real number and C be the curve given by: r(t) = m(cost – 2)ỉ – msintj + (mt –…

A:

Q: Find the length of the curve y = In(1 – x), 0sxs.

A: Given: y=ln(1−x2)0≤x≤12

Q: Prove that the following curves are intersecting with a right angle: 5y-2x+ y -x²y =0 2y+5x+ x* –…

A:

Q: Find the length of the curve. 30) x = 3 ty 3/2 - y 1/2 from x = 16 to x = 25

A:

Q: Use Green's theorem to evaluate ſ. (5xy + x² + y²) dx + (x² – y)dy where C is a closed curve that is…

A: We have to evaluate the integral ∫5xy+x2+y2dx+x2-ydy over the closed curve C. Comparing with…

Q: Let C be the curve y = x³ – 3x2 + 4x – 1 joining the points (1, 1) and (2, 3). Find the value of…

A: We need to find value of integral.

Q: Find the exact length of the curve.42. x = et − t, y = 4et/2 , 0 ≤ t ≤ 2

A: Given eqn : x=et-t,  y=4et/2 ,  0≤t≤2 So, we use dxdt=et-1,  dydt=2et2   Length of the curve =…

Q: Find the exact length of the curve. x = e' +", Osys 4

A: Given function is  x = e^y +(1/4)e^-y

Q: Examples Fid io av ed of the vegim borodkl by ke vegin Cuve and line hy mlars of double ifegrafin,…

A: Please refer the attached image for complete solution.

Q: Let C be the line segment connecting 0 to ni. Then f. 1-e22 dz = a. 0 b. -0.5 + ni C. πί d. None of…

A:

Q: The length of the curve y = (2+ 2x )3/2 between x=0 and x=2 is O 41/3 O 45/2 O 43/2 O 47/3 49/6 ООО…

A:

Q: 1. Find the exact length of the curve. (a) y² = 4(x+ 4)°, 00 (b) y = In z, 1<I<2

A:

Q: Trace the curve r = sinocos'o. 2)

A: Given: r=12sin3θcos3θ Explanation: r=12sinθcos3θat θ=0r=12sin0cos30=0⇒r=0at…

Q: |Verify that Green's Theorem is true for S,3(y2 – x) dx + (x² – y)dy given that the curve C is…

A:

Q: 3. For the curve x – xy² = 8e²4, do the following | a) Verify that the point (2,0) is on this curve.

A: Given, curve x3-xy2=8e2y

Q: Find the length of the curve r(t)=, -2 "< or equal to" 0, "< or equal to" 2.

A:

Q: (b) Verify that Green's Theorem is true for S,3(y² – x) dx + (x² – y)dy given that the curve C is…

A:

Q: 4. Find the are length of the curve from y=1 to y-4

A: The given curve is x=18y4+14y-2 from y=1 to y=4

Q: 3. Discuss how to evaluate f () dz , where C is a simple closed curve C=10 enclosing the origin.

A:

Q: Find the length of the curve y = a +; from a = 0 to x = 1. O None 5/2 O6/2 3 O4/2 3. 3

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Q: 99. Determine the concavity of the curve x= 2t + In t, y = 21 – In t. %3D

A:

Q: 1. Find the are length of the curve x = (y + 2)2 from y-0 to y=1

A: As per our guidelines we are supposed to answer only one question. Kindly repost other question as…

Q: b) Find the points on the curve 2 17x + 12xy + 8y“ = 100| that are farthest away from the origin…

A: Substitute x = rcosθ and y= rsinθ

Q: Let Q = 4x2 + 6xy+ 4y? . The points on the curve Q = 7 which are closest to the origin are: %3D %3D…

A:

Q: Use Green's theorem to find the k number that satisfies (k + 3) ∫C((k/2)*(x^k)*(y^2))dx + ((yx^(k +…

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Q: Let C be the curve connecting (0,0) to (9,3) to (-3,3) to (0,0) using straight lines. Evaluate (sec…

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Q: Find the exact length of the curve. y = In(1 - x²), o sxs

A: Find dy/dx, then put it in the arc length formula.

Vector Arithmetic

Vectors are those objects which have a magnitude along with the direction. In vector arithmetic, we will see how arithmetic operators like addition and multiplication are used on any two vectors. Arithmetic in basic means dealing with numbers. Here, magnitude means the length or the size of an object. The notation used is the arrow over the head of the vector indicating its direction.

Vector Calculus

Vector calculus is an important branch of mathematics and it relates two important branches of mathematics namely vector and calculus.