Show that the integral S. (3x²y + y)dx + (x3 + x)dy is independent of path, if it is, then evaluate it where c is any path between the points (1,2) and (4,5).

Q: Verify Green's Theorem by evaluating both integrals Lv² dx + x² dy dA ax ay for the given path. C:…

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Q: Use Green Theorem to evaluate C P xydx+e'dy where Cis the path from (0,0) to (1,1) along the graph…

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Q: Evaluate the line integral 2 Cx2y)dx- 2xdy along each of the following closed paths, taken…

A: Given that the integral,

Q: The set of all values of p which makes the integral x* dx is divergent is a) (-1, 0) b) (-n,-1) c)…

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Q: -(1,2,3) What should m be for the integral S00 (2xyz? + 3)dx + (mx²z³ – 2)dy + (3x²yz² + 4z)dz to be…

A: The curl of a conservative field, F→ is 0→ . In that case, the line integral of F→ is path…

Q: Find the line integral ye-xz ds Here, C is the path r(t) = [t,3t,-6t], 0sts In(8) ·

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Q: Let a, b> 0, then the integral x2a +1(1-x²)bdx, after changing variables y = 1– x will be expressed…

A: Given the integral I = ∫01x2a+11-x2 b dx To changing variables y = 1-x2 will be express

Q: Verify Green's Theorem by evaluating both integrals y2 dx + x² dy = dA ду for the given path. C:…

A: In this question we can verify the Green's Theorem . i.e, ∫C M dx +Ndy = ∫∫R ∂N∂x-∂M∂ydA .

Q: Use Green's Theorem to evaluate the line integral |(y - x)dx + (2x – y)dy for the given path. C: x =…

A: Let's find.

Q: Suppose F(x, y) = (6x + 5y)i + (5x + 4y)j. Evaluate the line integral for each of the given paths,…

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Q: Let o be the hypercube [0, 1]". For any f e C[0, 1] show that (). (x1 + x2 + + Xn dxjdx2 dxn = f ...…

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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: 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: Evaluate the integral ∮C 4x3y dx + x4 dy for any closed path C.

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Q: 1. Evaluate the line integral of E = xax - y ay along the segment P₁ to P2 of the circular path as…

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Q: Verify Green's Theorem by evaluating both integrals | y2 dx + x2 dy dA ду for the given path. C:…

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Q: 38. Evaluate the integral dy + x³ dx for any closed path C.

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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: Is dz

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Q: Evaluate the integral (6x-y)dx + (x+6y)dy along the path C, defined as y = 16-x² from (0, 4) (4, 0).…

A: Answer: The solution is given below:

Q: Verify Green's Theorem by evaluating both integrals v² dx + x² dy = / / dA ax for the given path. C:…

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Q: Verify Green's Theorem by evaluating both integrals y² dk + x² dy = / dA ay ax for the given path.…

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Q: 17. Integrate f(x, y, z) = (x + y + z)/(x² + y² + z²) over the path r(t) = ti + tj + tk, 0 < a < t<…

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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: 2. Evaluate the given line integral, (x – y, y) · dR, where C is the line segment from (0, 1) to…

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Q: Use Green's Theorem to evaluate the line integral (y - x)dx + (2x – y)dy for the given path. C: x =…

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Q: Evaluate the following line integral. x-ds; C is line segment from (3,5,6) to (- 3,12,2).

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Q: Verify Green's Theorem by evaluating both integrals aN dA Lv? dx + x² dy = ду ax for the given path.…

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Q: 15. Evaluate the line integral J. (3x²y* - 6xy ]dx + (4x²y³ - 3x°) dv, where C is any path from…

A: My=Nx means F is conservative vector field

Q: Evaluate the line integral | H. dř; Ħ = e" î + [xe" + In(z)] ĵ + 22 k, and C is the path defined by

A: Given that  H→=eyi^+xey+lnzj^+2yzk^      (1)  and r→=t,1,1, t∈0,1                (2)      from…

Q: Evaluate the line integral H. dř; Ħ = e" ì + [æe" + In(z)] ĵ +2² k, and C is the path defined by 7…

A: Given H→=eyi^+xey+lnzj^+2yzk^ We know that dr→=dxi^+dyj^+dzk^…

Q: Show that the function u = p(ry) + JFÿ¢(y/x) provides the quation „Pu = 0. dy

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Q: Verify Green's Theorem by evaluating both integrals dA [_v² dx + x² dy = [ ] (ON-OM) DA ду for the…

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Q: Let P denote the path r(t) = (7cost), where 0<t< 2n, and compute ſpF - dĩ where F = (!+ty 4 sin…

A: Let F→x,y be a vector field and r→t=xt,yt be the path of an object, where t1≤t≤t2. The line integral…

Q: 3. Evaluate the integral *zp- where C is the circle |2| = 2.

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Q: 2. Evaluate the given line integral, (-y, x – y) · dR, where C is the line segment from (2,0) to (0,…

A: Using parametric equation, line integration is evaluated.

Q: Use Green's Theorem to evaluate the line integral (y - x)dx + (2x - y)dy for the given path. C: x =…

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Q: Show that the integral is independent of the path, and use the Fundamental Theorem of Line Integrals…

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Q: Evaluate the line integral IF.då over the path C to Q (2,4) from P(0,) where F(xy) = 2xy2 + 3y f 131…

A: According to question, Given function, F→x, y = 2xy2 i^ +3y j^,We have to calculate line integral…

Q: Show that the following line integral is path independent an (3, 2) S cosh(+ X (0,0) (8xdx+6ydy)

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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: 3. Evaluate the integral o f(z) dz, where C is the unit circle enclosing the ori- gin, and f(z) = 1+…

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Q: Verify Green's Theorem by evaluating both integrals [ 1² dx + x² oy = √ √ (2x - 3) da dA dy əx JR…

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Q: Show that the following line integral is path independent and evaluate its value. Show the details…

A: We have  I=∫(0,0)(3,5) cosh(x23+y25)(10xdx+6ydy)apply Green's…

Q: Evaluate the given line integral. | (y, -x) dR, where C consists of 4y = x² from (-2, 1) to (2, 1)…

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Q: Verify Green's Theorem by evaluating both integrals ƏN [_y² dx + x² ay = [] (ax - 3+²) da d'A dy ay…

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Q: Verify Green's Theorem by evaluating both integrals aM | y? dx + x² dy = dA ax for the given path.…

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Q: 3. Evaluate the line integral / (x+ 2y)dx + x²dy, where C consists of the path C1 from (0,0) to (3,…

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Q: 37. Evaluate the integral fи 4x³y dx + x* dy for any closed path C.

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