Calculate . F · dr where F (x, y, z) = zi + xyj - y?k On the curve C r(t) = t?i + tj + vĩ k, 0

Q: Calculate the line integral f (VS) ·d7 for f(r, y) = -cos(2r-y)+ yr + y? and the curve C given…

A: Solution

Q: Evaluate f. x'dz +z?dy +y?dx where C is the curve x = t ,y = t2, z = t? from t = 0 to t = 2.

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Q: Show that at any point (x,y) on the curve y=bex/a ,the length of the subtangent is constant and the…

A: The given curve is y=bexa.

Q: Compute the line integral f 2xy dx + x2 dy along the following curves. (a) C1 along the circle x2 +…

A: A detailed solution is given in steps 2 and 3.

Q: Evaluate the line integral along the curve C. (xz+2) ds, C is the curve r(t) = (-6-2t) i + tj- 2tk,…

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Q: Find the tangent plane of the surface z = f(x, y) at (0,0, 0) in which 3y f(2, y) = || cos /3 t dt.

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Q: Evaluate fpxV47x² + 47y² dA, where D is the shaded region enclosed by the lemniscate curve r? =…

A: To find the value of the double integral         ∫∫Dx47x2+47y2dA where D is the region enclosed by…

Q: Evaluate the line integral along the curve C. (xz + y) ds, C is the curve r(t) = (-4 - t) i + 2t j -…

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Q: Evaluate the line integral for the given curve C. S-ds, C: r(t)= (t',t*),1<t<2 cy ||

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Q: Calculate the integral of f(x, y, z) = 7x² + 7y² + z³ over the curve c(t) = (cost, sin t, t) for 0…

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Q: Evaluate the line integral F(x, y) = xyi – y²j where the curve C is repre sented by R(t) =t²i+t³j…

A: Function : Fx, y = xy i - y2 jRt = t2i + t3j from (1,1) to (4, -8)

Q: Evaluate the line integral F dr where F (x,y) - xy'i+ yj and the curve C is the shortest path from…

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Q: Evaluate the line integral along the curve C. 1. 21 10 21 2. 20 21 3. 20 8 ds, C is the curve r(t) =…

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Q: Evaluate ∫c (y+sin x) dx + (z2 + cos y ) dy +x3 dz where C is the curve r(t) = (sin t, cos t, sin…

A: ∫C y+sin x dx + z2 + cos y dy + x3 dz = ∫C y+sin x i + z2 + cos y j + x3 k (dx i + dy j + dz k)…

Q: Find an equation for the level surface of the function f(xy.z) = x+ ey+z that passes through the…

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Q: Evaluate the line integral (xy dx + y dy), where C is the curve y = sin(x), 0 ≤ x ≤ π/2

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Q: Evaluate the line integral f, F . dr, where C is the curve given by F(t) = 3 cos ti+ 3 sin t j, 0<t<…

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Q: Evaluate the integral | (3x – 2 y + :)ds where C is the curve, C, and C, as shown. 3- y² + z² 2. 3…

A: Here, Curve C is equivalent to c1+c2. The point of intersections of c1 and c2 is at 0,1,0.

Q: Given the curve X = 2y – y and the line X = 0 Find the moment of R with respect to the y-axis M. y…

A: Given problem:- Given the curve x = 2y-y2 and the line x = 0 Find the moment of R with respect to…

Q: The value of the line integral | F(r).dr when C F(F) = [-y, -xy]and C is the upper half circle arc…

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Q: Let C'be the curve represented by the equations x=t, y=3t^2, z=6t^3, where te[0,1]. Evaluate the…

A: Given that the curve C has the equations as,  x=t, y=3t2, z=6t3  where t∈0,1 To evaluate the line…

Q: Suppose C is the curve r(t) = (6t,t°), for 0sts2, and F = (3x,3y). Evaluate F•Tds using the…

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Q: Evaluate the line integral ∫Cx2ds, where C is a the curve given by the equation y=f(x)=lnx, 1≤x≤e.

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Q: Let G(x. y)=x*yi+(y+xy*)j. Evaluate the line integral [G-dr, where Cis the curve x= ť and y=t* for…

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Q: Evaluate the integral c (xy+z^2)ds where c is the arc of the helix x=cos t ,y =sin t, z=t which…

A: To evaluate: The integral ∫c (xy+z2) ds where C is the arc of the helix x=cos t, y = sin t, z = t…

Q: Evaluate (y + 3 sin(x)) dx + (z2 + 9 cos(y)) dy + x³ dz where C is the curve r(t) = (sin(t), cos(t),…

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Q: Evaluate (y + 3 sin(x)) dx + (z2 + 4 cos(y)) dy + x³ dz where C is the curve r(t) = (sin(t), cos(t),…

A: Given: ∫Cy+3sin xdx+z2+4cos ydy+x3dz to evaluate. The parametric equation is rt=sint, cost, sin 2t,…

Q: Show F(x, y) = (4xy³ + 6)i + (6x²y² + 2e²¹)j LF is conservative by finding a potential function f…

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Q: Evaluate the line integral 2xyz dx + x²z dy + x?y dz where C is an oriented simple curve connecting…

A: Given line integral is,

Q: Calculate integral R C F dr · dr, where F(x, y, z) with a parameterization r(t) =, 1 , and C is a…

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Q: Evaluate the line integral F. dr, where F (x, y) = (3+2xy) i+ (x² – 3y²) j and C is the curve given…

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Q: Evaluate the line integral, where C is the given curve. ∫ c xyeyz dy, C: x= t, y= t2 , z = t3, 0…

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Q: 18. Integrate f(x, y, z) = – Vx² + z² over the circle r(t) = (a cos t)j + (a sin t)k, 0 <t< 2m.

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Q: 3. Evaluate the line integral c F. dr where F(x, y, z) = (e², xz, (x + y)) with 0 ≤ t ≤1 over the…

A: The solution is given as

Q: Evaluate the line integral, where C is the given curve. xy ds, C: x = t2, y = 2t, 0sts 3

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Q: Let F(x, y) = -z|. Let C be the curve parametrized by (t, cos(t), sin(t)), where 0 <t<. y Evaluate:…

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Q: 3. If Ø = 2xyz?,F = xyî – zĵ + x²k and C is the %3D curve x = t2, y = 2t, z = t³from t = 0 to t = %D…

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Q: Use Green's theorem to evaluate the line Integral for the curve C given in the figure. [2y dx + x…

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Q: Evaluate the line integral. y dx + z dy + x dz, r(t) = (2 + t-1, t3, t2) for 0 <t< 3

A: Here, x=2+t-1 y=t3 z=t2

Q: Civen tz f(x,y) = J 36-9x=y

A: If the graph of z=f(x,y) is sliced by a plane: z=c, then the two dimensional cross-section obtained…

Q: Evaluate the line integral, where C is the given curve. xy ds,     C: x = t2,    y = 2t,    0 ≤ t…

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Q: Show that the curve r(t) = (ln t)i + (t ln t)j + t k is tangent to the surface xz2 - yz + cos xy = 1…

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Q: Determine which of the following gives the area of the surface obtained by revolving the curve y =…

A: This question is related to surface area. We will solve this question by using formula of surface…

Q: Evaluate the line integral ſp(xi + Vyi) · dr where D is the oriented curve shown below. (1,1) y = x2…

A: We know that r=xi+yj As, the path followed is y=x2 So, r=xi+x2j Let x=t, then  r=ti+t2jdrdt=i+2tj…

Q: Evaluate the line integral 2xyz dx + x²z dy + x?y dz where C is an oriented simple curve connecting…

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Q: Evaluate the line integral. e ds where C is the curve given by x=2t, y=3t, 0<t<2

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Q: How many points are there on the surface z = dx +y such that the tangent plane to the surface is…

A: Given :  d=maxa,b,c⇒d=max2,1,0⇒d=2 So,z=dx2+y2⇒z=2x2+y2 Also, dx+y+z=0⇒2x+y+z=0 Let fx,…

Q: Find the area bounded by the line y=4 and the curve described parametrically by: x(t) =…

A: Given: y=4x(t)=t-1t,    y(t)=3t+1tSince y=4,  y(t)=3t+1t ⇒ 3t+1t=4⇒3t2-4t+1=0⇒3t2-3t-t+1=0…

Q: 3. Integrate f(x, y, z) = Vx² + z² over the circle (a cos t)j + (a sin t)k, r(t) r(t) 0 <t< 2m.

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