a bounded by the curve y = 3VI and the lines z = 6 and z= 8 above the z – azis is Võ+ 12v6 O 10

Q: Find the minimum distance from the point P with coordinates (1. 2. 1) to the line r = a+Ab, where a…

A: The minimum distance from the point P(1,2,1) to the line r is the perpendicular distance between P…

Q: Consider the curve segments: 1 - to x = 4 and 3 1 to x = $1: y = x² from x S2: y = Vx from x = = 16.

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Q: 1 - Findthe lenght of the Curve a) r(t)* = (3cost)i³ + (3sint)j* + 4tk¬ for 0 < t < 2n b) r(t)* =…

A: Use formula of length of curve

Q: 3) Find the exact length of the curve: x = y = In(1+t); 0<t<2.

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Q: Calculate the are bounded by the curves: C. y = e2x and x - 2y + 5 = 0

A: Introduction: The formula of the area between the curves y=f(x) and y=g(x) is given by,…

Q: 13- Find the length of the curve a) r(t) = 4costi + 4sintj³ + 3tk³ 0< t < 3n b) r(t) = 4ti + 4tj¬ +…

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Q: 1 Find the exact length of the curve: r(t) = , 6.

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Q: 2 32 |for 1st<3. length of the curve c(t) =| In t,-t"

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Q: The numerical value of 34 3dA (where D is the region bounded by lines y=0 and x = 1, D and the…

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Q: 3. Calculate the line integral S F - d7, where F = (x²y³, x³y?) and C is the path given by C 7(t) =…

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Q: The area bounded by the curve y3a and the lines z 4 and a-9 above the z-azis is aquare units O 27v0…

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Q: Evaluate Vf dr where f(x, y, z) = 4xye and C is the path parameterized by r(t) = costi+ sin tj+tant…

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Q: The area bounded by the curve y= 3va and the lines z = 3 and T = 8 above the z- aris is square units…

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Q: s Compute yz dz + zzdy + zydz along the curve (t, t2, t³), 0 st<1

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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: 1) Evaluate S,(x – y)dx + (y – z)dy + xdz where C is the curve from (1. –2, 3) to (–4, 5, 2) if a) C…

A: Given integral: ∫Cx-ydx+y-zdy+xdz Here, C is the curve from 1,-2,3 to -4,5,2 a) The parametric…

Q: find the Lineartransfromation T:Manz (R)R whose matrit Mp --

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Q: Show that the opelator Tilol defined by Y=19;)=Tx.-E] *=(h:)is Linear and bounded

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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…

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Q: Compute the length of the curve over the given interval. r(t) =〈3t, 4t − 3, 6t + 1〉, 0 ≤ t ≤ 3

A: Given rt=3t,4t−3,6t+1,  0≤t≤3Formula→Arc length=∫abr't dtr't=f't2+g't2+h't2    rt=xt,yt,zt

Q: What are the control points of the splines Bezier curve from the two equations given below: a(x) =…

A: Given, ax=1+5x2+2x3...1bx=1-x+x3...2 The control points are lie on both the curves. So, to get…

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: compute the length of the curve over the given interval.  r(t) = (2t2 + 1, 2t2 − 1, t3), 0 ≤ t ≤ 2

A: Length of a curve is also like computing distance between two points but with the restriction that…

Q: 38. Calculate y = x, x = 0, and x = 1. || e* dA, where D is bounded by the lines y =x +1,

A: We have to find ∫∫D ex dA, where D is bounded by the lines y=x+1, y=x, x=0 and x=1

Q: If lx + my = 1 touches the curve (ax)" + (by)" = 1, show that %3D )+) = 1.

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Q: If C is the line segment from (1,3) to, (0, 0), find the value of the line integral: S(3y² i +3x}) ·…

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

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Q: Find the value of c such that the regionbounded by y = c sin x and the x-axis on the interval [0, ∏]…

A: Given y = csinx and the region bounded by y = c sin x and the x-axis on the interval [0, π] has area…

Q: # 4. Evaluate the line integral | (x +2x²y) ds, where C is the upper half of the unit circle x2 + y2…

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

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Q: y6 = C (x² + 1) %3D

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Q: Given the one-parameter family of curves y² = Cx³ + 5. Find the family of orthogonal trajectories.…

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Q: a) Evaluate the line integral S. (2x – y) dx – (x + 3y) dy, where C is a straight line from (1,1) to…

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Q: 4. Evaluate S. y² – 3z + 2 ds for each of the following curve C is the line segment from (1,0,4) to…

A: We have ∫Cy2-3z+2 ds C is the line segment from 1,0,4 to 2,-1,1 Now, the parametric equation of the…

Q: The shaded area shown in (Figure 1) is bounded by y axis and the curve y? = (1.44 – x) m, where z is…

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Q: Let a curve be parameterized by x = t³ – 9t, y =t+3 for 1< t< 2. Set up (but do not evaluate) the…

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Q: Find the equations of the normal line to the surface z = 2a'y* at the point (2, – 1, 16). | x = y =…

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Q: 13 ABCD is a trapecium with ABDC. A line parallel to AC intersects AB at X at Y Prove that ar…

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

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

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Q: (b) F. dr, C is the line segment from (1, 1) to (5, 1), F = (y, -x). 3 of 7 (c) Set up but do not…

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Q: 2. Find the area of the smaller loop of the curve r = = 3( 1- 2cose)

A: We will solve the following

Q: Consider f(x, y) 39 x2-y?, and let z = f(x, y). What is the z = 9 level curve? O A circle of radius…

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Q: Show that the line integral / 2re" dx + (2y – x²e¯") dy is independent of path where C is any path -…

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Q: Consider Gollow Sx2 of Eunluate y when y to Find y Evalnate of when 11 Write Mhe etios to the Write…

A: Now when x = 1, Now differentiating given equation with respect to x, we get Now when x = 1,

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: Below is a curve C, which is the intersection between the surfaces  S1 : z = 4 − x2 S2 : z + 2x −…

A: Given that,Below is a curve C, which is the intersection between the surfaces   S1 : z=4−x2  S2 :…

Q: FInd c and d so that the curve y = x3+ cx2+d will have a relative extrema at (2,5). Find other…

A: Maxima or minima: Let f(x) is any function. 1) Evaluate f'(x) = 0 for critical point. 2) Evaluate…

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.