a) Write The Green's Theorem in the plane relating double integrals and line integrals. (Please, don't give examples and don't write your comments.)

Q: 1 = S I .3 x³ sin(2x) (x² + 1)² -dx.

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Q: Find the value of w = 5 bounded by y = 0, y = 2,0 ≤x≤ 2y, and x ≤z≤y + 4.

A: We have to find the value of w=5. It is bounded by:y=0, y=20≤x≤2yx≤z≤y+4

Q: +x-2xy) ds once around the circle x² + y² = 4, z = 0

A: ∮C(2+x−2xy)ds once around the circle x2+y2=4,z=0x2+y2=4Using parametric…

Q: 19.25. Let C be the boundary of a square whose sides lie along the lines x = ±3 and y +3. For the…

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Q: 1 Circles

A: To Setup: Integrals for the area of the shaded region in both figures.

Q: sin z dz, C (z+1)7 tan (z/2) ii) J. dz C Z-T/2

A: To Solve the above integral we use Cauchy Integral Formula.Given in Second Step.

Q: 0.8 0.6 0.4 0.2 of -0.2 -0.4 -0.6 0.5 1.5 2 5 A direction field for y'= -(2xy2 + y³) 2.5

A: Given differential equation is: y'=-2xy2+y3. We have to sketch the direction field for the given…

Q: (b) Evaluate the line integral dz along the simple closed contour C shown in the diagram. 1

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Q: In whatever way you see fit, evaluate the line integral L(10xy-3y+21x dx+ (15xy=+6xyz*)dy+ (15xy='…

A: given line integral ∫c10xy3z3+3y2z3+21x2z2dx+15x2y2z3+6xyz3dy+15x2y3z2+9xy2z2+14x3zdz   ...i…

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

A: Given integral is ∫Cy−xdx+2x−ydy where C:x=2cosθ,y=sinθ,0≤θ≤2π. Green theorem is given by…

Q: 2ху dx + (x + у) dy Jc C: boundary of the region lying between the graphs of y = 0 and y = 1 – x2

A: Given: ∫C2xy dx+(x+y)dy y=0 and y=1-x2

Q: rcos (y) dA, R= [-2, 3] × [0, ]. Evaluate R

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Q: 2.Proof that any tangent plane for the surface F(- point = 0 passses through a fixed

A: Given below clear explanation

Q: Evaluate the surface integral.

A: We need to find the surface integral of

Q: Evaluate the integral [] exp(-5x² − 2xy − 5y² + x)dxdy R2

A: The rule for a Normal density function is fx,μ,σ2=12πσ2e-x-μ22σ2 where the mean of the distribution…

Q: Define and explain integral and its properties briefly

A: Explanation of the answer is as follows

Q: Evaluate the line integral

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Q: Consider the integral o Sa -9-x² (x² + y²)dxdy. (a) Sketch the domain of integration. (b) Compute…

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Q: What is the value of the contour integral of a complex function over a closed curve?

A: The value of the contour integral of a complex function over a closed curve is zero if the function…