Calculate the integral of ƒ(x, y, z) = 2x² +2₁² + zº over the curve c(t) = (cos t, sin t, t) for 0 ≤ t ≤ ñ. Sc(2x² + 2y² + zº) ds =

Q: Show that / F dr is independent of its path, given that F(r, y, :) = + tan-:) i+tan rj+ k. IHence,…

A: Here we will find out the required value and prove the required result.

Q: 21 Let F(x, y) = (3x + 2y)i + (2x - y)j. Evaluate F. dr where C is the curve X y from (0, 0) to (1,…

A: Here we calculate the integral along the curve x = y2 from (0,0) to (1,1)

Q: Let C be the curve represented by the equations 5t, y = t, (0 < t< 1). In each part, evaluate the…

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Q: 63; Evaluate the întegral +3 dx + 6xy dy where Cay Curve Joininy (t, -) to (16, I),

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Q: r(4) (arcsin(t), arcos(1),0

A: Given: rt=arcsint, arccost, 0   As we know that; ddxarcsinx=11-x2ddxarccosx=-11-x2

Q: Find an equation of the tangent plane at P = (0, 3, – 1) to the sur- face with equation ze* + ez+1 =…

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Q: The value of the line integral S(2xy dx + 2x?ydy + dz) C along a path joining the origin and the…

A: We have to find the value of given line integral.

Q: F(x, y) = (4x³,3y²) along a path from (0, 0) to (2, 6). Find S F · dř. a) using a parabolic path. b)…

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Q: Show that the tangent at P(x,y,) the curve Vx+ Vy = Va is yy,² +xx, ² = a²

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Q: ntegrate ze²² dz where C from 1 along the axes to i C

A: (.)   Given integral is,    where  along to the axes to .(.) If a function is analytic then integral…

Q: Let C be the curve represented by the equations X = 4t, y = t², (0 ≤ t ≤ 1). In each part, evaluate…

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Q: What is the integral of the constant function f(x, y) = 5 over the rectangle [-2, 3] x [2, 4]?

A: Given:            f(x, y) = 5   over rectangle -2, 3×2, 4

Q: 6.(6 pts) Evaluate the line integral / (x+ y + 2) ds along the curve C given by r(t) = (sin t, cos…

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Q: Let f(x, y) = 6x – 2ry and the point M(5, 9). Among all unit directions, there is a unit direction…

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Q: Let f(x,y)= e, Compute the cartesian equation of the tangent plane r,y) at the point (2.2).

A: The equation of tangent plane

Q: F. dr where F = yzi + xzj +xyk and C is the curve x = t, y = 2t², z = –2t² with 0<t < 1.

A: Given F=yzi+xzj+xyk and r=ti+2t2j-2t2k Now the vector field along the curve can be given by…

Q: Consider the function f (x) = cos (2x) a) Find the polynomial of degree 4 centered at X = 1.…

A: The given function is fx=cos2x

Q: Let the curve C be the line segment from (0, 0) to (3, 1). Let F = ⟨2x-y, 4y-x⟩ Calculate the…

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Q: | 5y? dx +xy dy

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Q: Calculate the tangent plane for the function f(x, y, z) = ecos(x)+sin(y+z) at the point (7,0,0).

A: To calculate the tangent plane for the function  at the point .

Q: Definition 3.27 Let P(x, y) be a point in R² and 0 be the radian measure of the angle formed by OP…

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Q: Find an equation (x(t), y(t), z(t)) of the line that is the intersection of tangent planes at (1,…

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Q: (il) Find the point in which the tangent line to the curve F-12-30)1+ 2+Int'+ k at t=1 intersects…

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