(6) Evaluate the line integral / r²ydx + xy°dy, where C is the square with vertices (0,0), (1,0), (1, 1) and (0, 1) by two methods, (i) Using direct method, and (ii) Using Green's Theorem.

Q: 5. Verify Green's Theorem for 0(xy +x )dx+(4x-1) dy where Cis shown below by (a) computing the line…

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Q: EXAMPLE 2 to evaluate the Use the change of variables x = u² - v², y = auv integral SSR y dA, where…

A: The objective is to evaluate the integral using change of variable. x=u2-v2, y=2uvy2=4-4x, y2=4+4x,…

Q: Find the solution in the form of Fourier integrals: U4-3uzz = 0, |u(x, t) bounded as x → ±0, sin r…

A: We are given the following second order partial differential equation : ut - 3uxx = 0  ;  -∞ < x…

Q: evaluate both the line integral and the double integral. 16. P(x, y) = 2x – x³y°, Q(x, y) = x³y°, C…

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Q: Use Green's Theorem to evaluate the line integral (y - x)dx + (2x - y)dy

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Q: Use Green's Theorem to evaluate the line integral along the given positively oriented curve. xy2 dx…

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Q: 6. Evaluate the line integral g. x²y² dx + xy dy where C consists of the arc of the parabola y = x²…

A: (a) Direct method The given line integral is ∮C x2 y2 dx + xy dy. The curve C consists of    C1 :…

Q: [[(2x + y²) da + 2xy dy], where C is the line segment from (-1,0) to (1, 2) Evaluate in two…

A: The line integral ∫CMdx+Ndy involves the integration over a specific path. In this case, the line…

Q: Plot a direction field and some integral curves for xy' - 2y = -1 (A) on the rectangular region {-1…

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Q: (2) Find the work done by F(x, y) = (2xy³, 4x²y²) moving a particle on the path C where C is the…

A: Formula: Work done W by a force field F on a particle on the path C is: W=∫C F·dr It is given that…

Q: Use Green's Theorem to evaluate the line integral along the given positively oriented curve.  xy2…

A: Our Aim is to evaluate the line integral along the given positively oriented curve xy2 dx+3x2y dy…

Q: 2. Use Green's theorem in order to compute the line integral P (x – 4)³ dy – y³ dx where is the…

A: We need to compute the given integral using Green's theorem . ∮C(x-4)3dy-y3dx Where is the circle of…

Q: 3) S. (3x² – 3yz + 2xz)dx + (3y² – 3xz + z²)dy + (3z² – 3xy + x² + 2yz)dz - Calculate the…

A: Let's find.

Q: Calculate the double integral (x + y)²ex* - y² dx dy where R is the square with vertices (2, 0), (0,…

A: According to the given information, it is required to find the integral over R where R is square.

Q: 16. Let D-((, y, z):05zs2, 0≤ y ≤1, 05:5z+y). Find the triple integral: 72xy³ dv.

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Q: Evaluate the line integral (direct method) | (3x– y) dx – x dy , where C consist of the parabola…

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Q: Evaluate the triple integral xy dV where E is the solid tetrahedon with vertices (0, 0,0), (6, 0,…

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Q: 1. Evaluate the line integral f(z, y) ds where f(x, y) = Vx+ 2y and C is the straight-line segment…

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Q: 8. Verify Green's Theorem for + (6y – 3y +z) dz + yx°dy where C is shown below by (a) computing the…

A: Let F=Px,yi+Qx,yj be the vector field and C be the boundary of the region D. The Green's theorem…

Q: Use Green's Theorem to evaluate the line integral S(x² + y²)dx + (x² – y²) dy where C is the…

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

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Q: Evaluate the line integral ∫AB(x+y)dx+xdy for two paths of integration: AB is the line segment from…

A: We need to find the following integral: a) AB is line segment from A(0,0) to B(1,1) b) AB is the…

Q: Evaluate the line integral (direct method) |(3x– y) dx – x dy , where C consist of the parabola y=…

A: Given integral is, ∫C(3x-y)dx-xdy

Q: Use Green's Theorem to evaluate the line integral along the given positively oriented curve. xy2 dx…

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Q: Consider the line integral xy dx + xy° dy with C the triangle with vertices (0, 0). (1,0), and (1,…

A: Given, the line integral Where C is the triangle with vertices.

Q: Find the value of the line integral 6xy dx + (3x² + 4y2)dy, where C is an ellipse 16 + = 1 from (4,…

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Q: How do I set up the triple integral of the function xy2 -3z, where the solid is bounded by the…

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Q: 5) Evaluate the line integral S YX²dX + XY²dY on curve C, where C is the upper portion of the www m…

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Q: |Evaluate the line integral: f.(r² + y) dx + (4x – cos(y)) dy where C is the boundary of the square…

A: According to Greens theorem, the line integral ∮CPdx+Qdy=∬DQx-PydA. BY the extended version of…

Q: Use Green's Theorem to evaluate the line integral. y dx + 7x dy C: square with vertices (0, 0), (0,…

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Q: Evaluate the triple integral. SSLĘ 8x dV x = 2y2 + 2z2 and the plane x = 2. where E is bounded by…

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Q: 8. Evaluate the line integral x dx + xedy where C is the closed path along the triangle in the…

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Q: 2/ Find the line integrals of the ellipse+3= ellipse by x = acos(t) ,y= bsin(t) 1 , parameterize the…

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Q: • Example 4 Evaluate the double integral /| v²x dA R over the rectangle R = {(x, y) : –3 < x < 2, 0…

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Q: 8. Verify Green's Theorem for (6y – 3y? +x) dx + yx*dy where C is shown below by (a) computing the…

A: Given integral is ∮C6y-3y2+xdx+yx3dy Here C consists of two line,  C1:Line joining 0,0 and…

Q: 6. Use Green's Theorem to evaluate the line integral fc 2xy dx + (x² + y²) dy, if C is the ellipse…

A: Let R be simply connected region with a piecewise  smooth boundary  'C'. If functions 'M' and 'N'…

Q: need to find the limits of integration in terms of ρ, φ, and θ.

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Q: Evaluate fff 12y - 8x dV wherein the limits of integration are the region behind y = 10-2z and in…

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Q: Evaluate the triple integral. 8x dV, where E is bounded by the paraboloid x = 2y2 + 2z2 and the…

A: The objective is to evaluate the triple integral ∫∫∫E8xdV, where E is bounded by the paraboloid…

Q: Use the Fundamental Theorem of Line Integrals to find the exact value of the path-independent line…

A: First of all hare we check for the path independent line integral then solve for moving over the…

Q: Using Green's theorem, convert the line integral §(6y² dx + 2xdy) to a double integral, where C is…

A: Given integral, ∮C6y2dx+2xdy Here C is the boundary of the square with vertices ±2,2 and ±2,-2.…

Q: The line integral (x+/y) ds , where C is the line segment joining the points (0, 0) and (1, 1) , can…

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Q: Evaluate the line integral: f y² dx + x² dy by parameterizing C where the curve C is a square R with…

A: According to the given information, it is required to evaluate the line integral.

Q: 4x Estimate the error in the quadratic approximation of f(x,y) = e sin y at the origin over the…

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Q: Evaluate the triple integral. 10x dv, where E is bounded by the paraboloid x = 2y² + 2z2 and the…

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Q: 4) Using Green's theorem, convert the line integral f(6y² dx + 2xdy) to a double integral, where C…

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Q: Use Green's Theorem to evaluate the line integral , 6y² dx + 5x² dy, where Cis the boundary of the…

A: Green's Theorem: ∮C(Pdx+Qdy)=∫∫R∂Q∂x-∂P∂ydA

Q: 5) Using Green's theorem, convert the line integral f.(6y² dx + 2xdy) to a double integral, where C…

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Q: 8. Evaluate the line integral (x'dx+ xe"dy where C is the closed path along the triangle in the…

A: Here we have to evaluate the integral ∫cx2dx+xe-ydy where C is the closed path along the triangle in…

Q: b) Evaluate [[ (3-4x)dV where E is the region below :=4-xy and above the region in the xy -plane…

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