Evaluate the line integral G. dr with G(x, y) = xyi+ 2y² j, and C is the closed path comprising the straight line segment from (1,0) to (0, –1) and three quarters of the clockwise circle of unit radius, centred at the origin, connecting (0, –1) to (1,0).

Q: Evaluate the integral f. x'y°dy – x³y?dx, where C is the square determined by the 0, y = 0, x =…

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Q: Evaluate the line integral I of the one-form w = (y+ z) dx + (x + z) dy + 10z dz over the line…

A: We have to find the line integral.

Q: Calculate the integral f, (3y – 3x²) dæ + (9x) dy where y= aD is the counter-clockwise boundary of…

A: we are allow to solve one question at a time thank you

Q: Given that C = C₁ U C₂ U C3, where C₁ is the line segment from (−1, −1) to (0, 0), C₂ is the line…

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Q: Evaluate : //| cos(x² +y³) dV, where E is the region bounded by the disk x² + y? < 9 E and the plane…

A: From the region of integration which is bounded by x2+y2≤9 and z=1, identify the limits of x, y and…

Q: Let C: r (t), te [a, b], be a regular curve in R°, oriented from the point P =r (a), to the point Q…

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Q: Evaluate the integral x² dx + y? dy where C consists of the arc of the circle x2 + y² = 4 from (2,0)…

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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: Let C: r(t), te [a,b], be a regular curve in R³, oriented from the point P = r(a), up to the F-dr…

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Q: Compute the line integral ∫_C F · dr, where F (x, y) = 〈−x^3, xy〉 and C is part of circle x^2 + y^2…

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Q: Can you show the steps on how to do this integral?

A: Consider the provided question, (3) Evaluate the integral by using the greens theorem,

Q: Evaluate the surface integral I F. dS when F(x, y, 2) = y²i+ 3x² j and S is the graph of - ža (1+æ²…

A: Here, Fx,y,z=y2i+3x2jz=121+x2-y2

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: Let f(x,y) = x 3 + y 4  + sin(2x+3y) . Determine the line integral of f(x,y) with respect to arc…

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Q: b. Evaluate the line integral ycos(xy)dx +(xcos(xy)+x)dy where R is the circle x'+y =9 traversed…

A: Solution Given b). ∫y cos(xy)dx+(xcos(xy)+x3) dyevaluate the line integral

Q: Evaluate the line integral xyld x + (x²+y)dy] +. where c is the saruare farmed by the lines

A: Let us consider a line integral ∫CP dx+Qdy then it can be written as  ∫D∫∂Q∂x-∂P∂y dA  Where, P and…

Q: Evaluate the line integral using Green's Theorem and check the answer by evaluating it directly.…

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Q: Evaluate the line integral I of the one-form w = xy dx + 12y dy over the parametric curve a : [0,…

A: Given: ω=xydx+12ydy To find: I=∫αω  where α:[0,ln2]→ℝ2 with α(t)=(e2t,e-2t) ∴x=e2t,y=e-2t Now,…

Q: Compute the line integral -xdy + ydx 2(x-y)2 + 4xy where L is the circle (1-x)2 + y? 3 with a…

A: We will find parametric equation of L and then convert whole equation in parametric form

Q: Let f(x,y) = 3x 2 y + e x+2xy . Determine the line integral of f (x,y) with respect to arc length…

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Q: Evaluate the line integral, where C is the given curve. xy ds, C is the right half of the circle x2…

A: C is the right half of the circle x2+y2=16 thus the parameterization of the circle is x=4cos t and…

Q: Evaluate the line integral, where C is the given curve. Lexy² ds, C is the right half of the circle…

A: Solution: I=∫cxy4ds

Q: Use either form of Green's Theorem to evaluate the following line integral. $9x²1² 6 C $9x³y² dx +…

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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: Evaluate the line integral along the given paths. y?) ds + (a) C: line segment from (0, 0) to (8,…

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Q: Use Green's Theorem to evaluate the following line integral. Of dy -g dx, where (fg) = (7x2,8y?) and…

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Q: Q7. Use Green's theorem to evaluate fy² dx + 3xy dy, where c is the boundary of the semi annular…

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Q: -) Let -y M(x, y) = x2 and N(x, y) = + y? x2 + y? how that ƏN ƏM dx ду ) Show that -y F(x, y) = x² +…

A: Given :-  A vector field for conservative field in the plane ,  ∂N∂x = ∂M∂y To Determine :-  M x,y =…

Q: Use Green's Theorem to evaluate the following line integral. Of dy -g dx, where (f,9) = (7x?,8y) and…

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Q: Evaluate the triple integral. SSS 5x²e" dv I where E is bounded by the parabolic cylinder z = 1 - y²…

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Q: Evaluate the line integral f. 4y dx + z² dy + (16 – x²) dz over the quarter of the circle of radius…

A: To Determine: Evaluate the line integral over the quarter of the circle of radius 4 in the xz-plane…

Q: Use either form of Green's Theorem to evaluate the following line integral. 03x y dx+ 4x y dy; C is…

A: According to the given information, it is required to find the value of integral using Green’s…

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: Given that C = C₁ U C₂ U C3, where C₁ is the line segment from (-1, −1) to (0, 0), C₂ is the line…

A: This is a problem on Green's Theorem. The problem has been solved step by step. See the solution.

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: Evaluate the integral | xy ds, where C is the curve consisting of the portion of the unit circle of…

A: Let us assume thatC1:x2+y2=1Put x=t,y=1-t2 and z=0.Hence,dx=dtdy=-t1-t2dz=0and 0≤t≤1.Let C2:The line…

Q: Let R:x²<ys 3 be the closed region having boundary C with counter-clockwise orientation. F=[X…

A: We are given that the closed region R: x2 ≤ y ≤ x/3, is having boundary C with counter-clockwise…

Q: 6. Let D be the region bounded by the parabola y = x² and the line y = 1. Let I be the boundary of…

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Q: Evaluate the line integral, where C is the given plane curve. xy ds, C is the right half of the…

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Q: Use the Green's Theorem to evaluate the line integral f, FdT where F = (sin(x²)-2x²y) i+ (cos(y?) +…

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Q: Evaluate the line integral, where C is the given curve. z? dx + x2 dy + y² dz, C is the line segment…

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Q: Let M(x, y) = y - cos y and N(x, y) = x sin y. Use the Green's Theorem to calculate the line…

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Q: Evaluate the line integral, where C is the given curve. Jo xy ds, C is the right half of the circle…

A: Solved below.

Q: Use Green's Theorem to evaluate the following line integral. Of dy -g dx, where (f.g) = (7x²,8y?)…

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Q: Evaluate the line integral Levis where C is the half circle x² + y² = 1, y ≥ 0. x² + y² ds

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Q: Suppose F(x,y)=(4x+6y)i +(6x+4y)j. Evaluate the line integral for each of the given paths, which are…

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Q: Consider the following line integral. xy dx + x?y dy, Cis counterclockwise around the triangle with…

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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: Find the line integral of F(x, y, z) = x'i + yj+ zk along the line segment C from the origin to the…

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