Use Green's Theorem to calculate 4. [(x² – xy) dx + (y² - 2xy) dy] where C is a square with vertices (0,0), (0,2) (2,2), (2,0).

Q: xxS y dS S is the part of the paraboloid y= x 2 + z2 that lies inside the cylinder x 2 + z2 =4

A: And the projection of S onto the xz-plane is given by the disk So using the parameterization in x…

Q: 2. Use Green's Theorem (and only Green's Theorem) to evaluate |(2? + y)dx + (x² – y³)dy, where C is…

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Q: Find a parametrization for the cylinder x2 + y2 = 1.

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Q: |Use Green's Theorem to evaluate [(3x² +y)dx+4xydy where C is the triangular C region with vertices…

A: The solution is given as

Q: x y 11. Use Green's theorem to evaluate O In(1+ y) dx –- dy , where C is the triangle with vertices…

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Q: Use Green's Theorem to compute y dx+(x³+3xy²) dy where C is the closed curve around the triangle…

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Q: Q4) Using Green's theorem to evaluate: (12 marks) Sc [ (x3 – xy³)dx + (y³ – 2xy) dy] Where C is the…

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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: Find c > 0 c > 0 such that the area of the region enclosed by the parabolas y = x 2 − c 2 y = x 2 -…

A: To find the value of c given area between the curves.

Q: 28. Evaluate by Green's theorem (x - cosh y) dx + (y + sinx) dy, where C is the rectangle with…

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Q: EXAMPLE 2 Find the area of the part of the paraboloid z = x + y' that lies under the plane

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Q: 4. Where does the curve r(t) = ti + t²j+ (3 t2)k intersect the paraboloid z = x2 +y² ?

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Q: Study the analytically of w = L(2) at the points z = 0, z= v2,z = -21

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Q: I need help finding the absolute maximum and minimum values of f on the set D.

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Q: The parametric form of the solutions of the PDES Uy + u u z = 0 , u(x,0) = -x is

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Q: 10. Use Green's Theorem to evaluate the integral fe y dx + 2xy³ dy, where C is the square with…

A: Introduction: By applying Green's theorem, one can determine line integral through surface integral.…

Q: Find F. dr, where F(x, y) = (xy² + cos x)i+(x²y+sin y)j and C is the ellipse C x2 + 2y2 = 9,…

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Q: 17. Integrate G(x, y, z) = xyz over the triangular surface with vertices (1, 0, 0), (0, 2, 0), and…

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Q: find the decomposition of a(t) into tangential and normal components at the point indicated, as in…

A: Consider the given information rt=eti+1-tj ,  t=0   Find the first derivative of rt…

Q: Study the analytically of w = L_r(z) at the points z = 0, z = v2,z = -2i

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Q: Find the linearization L(x,y) of the function f(x,y) = e 3× cos (9y) at the points (0,0) and 0,

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Q: 2. Evaluate f, z³ /x² + y² + z² dV where E is the solid hemisphere that lies above the xy-plane and…

A: Solution:- equation of sphere having centre origin (0,0) and radius=1 x2+y2+z2=1…

Q: Find the absolute maximum and minimum values of f(x, y) = 3xy−6x−3y + 7 on the closed triangular…

A: Find out the absolute maximum and minimum values of  f(x,y)=3xy-6x-3y+7 for the maximum and minimum…

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: • SSS ²³² x² + y² = 2, and the xy plane. Evaluate e² dV where E is enclosed by the paraboloid z = 6…

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Q: Find the absolute maximum and minimum of f(x,y)= 4xy^2 - (x^2)(y^2) - xy^3 on the closed triangular…

A: f(x,y)=4xy2-x2y2-xy3fx=4y2-2xy2-y3fx=04y2-2xy2-y3=04-2x-y=0…

Q: 3. Evaluate the line integral y² dx + x²y dy where C' is the rectangle with vertices (0,0), (5,0),…

A: Solution is below

Q: Use Greens Theorem to evaluate: §. (2x + ev´)dy - (4y2 + e*")dx where C is the boundary of the…

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Q: Evaluate the triple Integral. 3xyz dV, where T

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Q: Sketch the surfaces HYPERBOLOIDS (y2/4) - (x2/4) - z2 = 1

A: we have to sketch the surface of the hyperboloids  y24-x24-z2=1

Q: Consider D to be the region below the paraboloid z = 3+x² + y² , inside the cylinder x + y² =1 and…

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Q: 11. Evaluate ([(y+z³ )dS where S is the part of the paraboloid x = 4- y – z² that lies in front of…

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Q: Use Green's Theorem to evaluate | (v? – tan-1 x²)dx + (3xy + 4x – In (y2 + 4))dy where C is the…

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Q: Obtain the Solution x, y > 0, xy > 1 of 1 the PDE Ury Such that x+y 2у и 3 0, их on the hyperbola…

A: 1st we set up the initial value condition and draw the figure than set up PDE and solve it

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: 14. Under the paraboloid z=x² +y2 and above the disk x +y² <9 for (x,y) in the first quadrant.

A: we have to find the volume

Q: Evaluate p(10ey – 9x)dy – (yx – 5 In x)dx by using Green's Theorem where C is a trapezium with…

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Q: Evaluate € fff.₂²². e² dV where E is enclosed by the paraboloid z = 3 + z² + y², the cylinder z² +…

A: We need to find integral.

Q: Evaluate the triple integral.  (x2 + y2) dx dy dz; W is the pyramid with top vertex at (0, 0, 2)…

A: The given triple integral is,

Q: 1. Use a change of variables to evaluate || ソーx dA where R is the trapezoid with vertices (1,0,…

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Q: Evaluate the line integral y² dx + x²y dy where C' is the rectangle with vertices (0,0), (5,0), (5,…

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Q: Suppose F(x, y) = (2 + 5y, 6z – 5y²). Use Green's Theorem to calculate the circulation of F around…

A: Given : vector field F→x, y=x2+5y, 6x-5y2 and triangle C oriented counter-clockwise with vertices…

Q: (1) Find the area of the curved surface of the upper half of the sphere x2 + y² + z² = a²(a > 0) on…

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Q: find ⌠⌠F*dS if F(x,y,z)=2xi+2yj+2zk and S is the paraboloid z=8-x^2-y^2 lying above the rectangle…

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Q: Show that the graph of ř = (t sin t, t cos t, t²) lies on the paraboloid z = x? + y?.

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Q: 3.use Greens theorem s (1+x5 dx + 2xy dy ,where C is triangle With vertices (0,0),(1,0),(1,3)…

A: Apply greens theorem

Q: Find the average haight of the Paraboloid z- xry* over the square osx <2, osd<2

A: We need to find the average height of paraboloid

Q: 1. The accompanying figure shows two polygonal paths in space joining the origin to the point (1, 1,…

A: Given: The function with two polygonal paths in space joining the origin to the point (1, 1, 1).…