Evaluate c (xy + y + z) ds along the curve r(t) = 2ti + tj + (4 – 2t)k, 0st<1.

Q: Integrate F(r)= [e2*, e²y,e²²] along the helix that cuts the z-axis at c.

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Q: On what interval is the curve t2 X. dt y% = Jo t2 + t +2 concave downward?

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Q: find the work done by F in moving a particle once counterclockwise around the given curve. F = 2xy3i…

A: Consider F = 2xy3i + 4x2y2j We have M = 2xy3 and N = 4x2y2

Q: (B) The point on the curve r(t)=Dti + (1 + r)j at which r(t) and r'(t) are perpendicular is (1,0).

A: This question is about application of vector calculas

Q: b) Evaluate S(2y + 4z)dx + (2æ + 3z)dy + (4x + 3y)dz, where C is the curve given by r(t) = for 0 <t…

A: given ∫C2y+4zdx+2x+3zdy+4x+3ydz where C is the curve given by rt=t3,2sinπt2,3cosπt2 for 0≤t≤1…

Q: find the work done by F in moving a particle once counterclockwise around the given curve. F = (4x -…

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Q: Q:1) Evaluate the line integral: [[xy'dx+x²ydy] C: [(x, y): x=t₁y=f²,0 ≤t≤1].

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Q: Find the length of the curve y = Vcos 2t dt from x = 0 to x = T/4.

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Q: Find dy/dx and d²y/dx² at t = 2 for the parametric curve x = ½t², y = ⅓t ³.

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Q: Find the linearization L(x,y) of the function f(x,y) = e (X cos (5y) at the points (0,0) and 0,- 2

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Q: Find the path of a heat-seeking particle placed at point P on a metal plate whose temperature at (x,…

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Q: Use Green's theorem to evaluate (2y dx-x dy) around the closed curve C. 27 (2,8) 8. 4. C. -2 (1,-1)

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Q: Find the work done by F = 2xy°i + 9x<y*j in moving a particle once counterclockwise around the curve…

A: The solution is given below:

Q: dy Q1) Compute for y(x) = sec (2x) + 2x-1 and dt x(t) = 4cos(t2) at t-1 %3D

A: From the given problem:   yx=sec32x+2x-1xt=4cost2   Now applying the chain rule:   dydt=dydx.dxdt…

Q: Express the curve by an equation in x and y given x(t) = sin(t) and y(t) = 8 + cos² (t). a) O-x² +y…

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Q: Compute I = (2x²132²y+ 4xy)ax +(x'+2x²-y') dy+zdz along The curve C C4U Cz UG given in the figure…

A: Given To compute I=∫C2x3+3x2y+4xy.dx+x3+2x2-y2dy+zdz along the curve C=C1∪C2∪C3 given in the figure.

Q: For the curve y = f(x) = x², 0 ≤ x ≤ 2, its length is 2 L = 1+ dy dx dx =

A: Given curve y=f(x)=x2,0≤x≤2 The objective is to find the length of the curve.

Q: B) Evaluate the line integral F-dr. where F (x, y) = x*y²i+x²y°j and C is the curve r (t) = (t³ +…

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Q: On what interval is the curve y = 0, v- [ t2 dt t2 +t + 2 concave downward?

A: NOTE: Refresh your page if you can't see any equations. . to find the inflection point, first, we…

Q: 1. Evaluate S. (2x+ y² – z) ds where C is the piecewise smooth curve below. (0, 1, 1) (0, 0, 0). (0,…

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Q: 6. (a) Show that F = (3a²y+2, x3 + 3) is conservative. (b) Evaluate Sc(3x²y +2)dx + (x³ + 3)dy where…

A: According to the problem, we have

Q: Integrate f(x,y) = x-5y over the curve C: x +y = 64 in the first quadrant from (0,8) to (V32,/32).…

A: Formula Used:Line integral ∫Cf(x, y) ds=∫abf(x(t), y(t)) dxdt2+dydt2dt

Q: Consider the closed curve in the xy-plane given by x² + 2x + y* + 4y = 5. dy -(x+1) (a) Show that dx…

A: Given data:   The expression for the given function is x2+2x+y4+4y=5.   (a) Differentiate the above…

Q: Q) Compute for y(x) sec (2x) + 2x -1 and x(t) = 4cos(t) at t-1

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Q: Q1: Use Green's theorem to evalute f. (xy + y2)dx + x²dy by the curve y = x and y = x2.

A: Given problem:- Use Green's theorem to evaluate ∫c (xy + y2 )dx + x²dy  by the curve y = x and y =…

Q: Evaluate S. (2xy – x²)dx + (x + y²) dy such that c is a curve bounded by y? – x , y – x² ?

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Q: %3D y"(t) = 4y(t) – e-tU (t – 1), y(0) = y'(0) = 0.

A: Solution

Q: Evaluate the line integral. { 4) ( y2 dx + x dy; C is the curve x =t2 - 1, y = 4t, 0 st s 1

A: The given line integral is ∫Cy2 dx + x dy; C is the curve x=t2-1, y=4t, 0≤t≤1

Q: Integrate f(x,y)=x²-7y over the curve C: x² + y² = 64 in the first quadrant from (0,8) to (√32.√32)…

A: Draw the given equation of the curve x2+y2=64 and the given points. The given curve is between…

Q: 2. Find an equation for the plane tangent to the surface 3x2 - 4xy+ z2 = 0 at the point (1, 1, 1).…

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Q: Use the method of Example 4 to compute dy/dx p at P = (2, 1) on the curve y^2x^3 + y3^x^4 −…

A: we can solve by this implicit differentiation, so,

Q: e the line integral Sc(3x?y+2)dx+ (x+3)dy where C is the curve r(t) = (t2, 3+t²) with 0 <t < 1.

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Q: B) Evaluate the line integral F-dr, where F (x, y) = x²y³i+ x°y°j and C is the curve r (t) = (t³ –…

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Q: Find the linearization L(x,y) of the function f(x,y) = e* cos (9y) at the points (0,0) and ...

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Q: Evaluate the line integral: [ y dx + x dy ; C is the curve x 1 = t² - 1, y = 6t, 0 < t s 1

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

Q: (A) Evaluate" [(x² + y)dx + (x + 1) 1+ along the path ABCA shown in figure C(0,3) A(0,1) x=3sint…

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Q: (2² + y² – 1)³ – x²y3 = 0, 2- Given the curve dy find dx -2 2

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Q: Evaluate § F · ds, where F = 2x(x+y)i + (x² + xy + y?)j and the positively-oriented curve C encloses…

A: Here we will use greens theorem to find line integral of vector field For a vector field F→= Pi^+Qj^…

Q: xe' dx + x² ydy 3. ; Compute along the curve x = 3t, y = t ,0St<1 С:

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Q: 2. If F(x, y, z) = y²i+(2xy+e3=)j+3ye³=k. Evaluate F·Î ds, where C is the curve represented by r(t)…

A: Given F→x,y,z=y2i^+2xy+e3zj^+3ye3zk^ and the curve is C given by r^t=cost,t,sint,  0≤t≤π2. Find…

Q: Show that the curve r(t) = ( t3 /4 - 2)i +(4/t-3)j+cos(t-2)k is tangent to the surface x3 + y3 + z3…

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Q: Find the path of a heat-seeking particle placed at point P on a metal plate whose temperature at (x,…

A: Consider the given function, Tx,y=100-x2-2y2 the differentiate partially, ∇T=-2x, -4y the slope at…

Q: On what interval is the curve t2 dt t2 +t+2 y = concave downward?

A: The given curve is: y=∫0xt2t2+t+2dt Using fundamental theorem of Calculus says: ddx∫axf(t)dt=f(x)…

Q: Compute ∫C y dx + xy dy if C goes from (0, 0) to (1, 2) along the line parameterized as x = t, y =…

A: Here I am attaching image so that you understand each and every step.

Q: Q1: Use Green's theorem to evalute S. (xy + y2)dx + x²dy by the curve y y = x2.

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