[Find div(F x G) and Curl (F x G) where F(x,y, z) = x²yi + y³z²j – 2z²k %3D and G(x, y, z) = 3xyi – 5yzj + 4z²k at (-1,-1,1). 1 %3D

Q: 2. Give iterated integrals for (xz*)dS where S is the part of the cylinder x² + y = 9 above the…

A: Given that x2+y2=9 and the part of the cylinder which is above XY plane implying that z > 0. We…

Q: 9) Use Fubini's Theorem to compute the double integral ƒƒ_ƒ(x,y)dª_where…

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Q: Evaluate the double line integral S  of the fuinction (x − 2y + z) dS. S: z = 9 − x,  0 ≤ x ≤ 9,  0…

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Q: Evaluate the double integral I2 = || v?dzdy T where T is a triangle with vertices (-2, 0), (0, 3),…

A: Here we have to evaluate the given double integral where T is a triangle with vertices…

Q: Evaluate the triple integral. 2xy dV, where E is bounded by the parabolic cylinders y = x² and x =…

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Q: Evaluate the line integral f e-*(cos y dx - sin y dy), where c is the rectangle with vertices (0,0),…

A: The line integral function is ∫Ce-xcosydx-sinydy. The vertices of the rectangle is 0, 0, π, 0, π,…

Q: Use symmetry to evaluate the double integral. 4xy 1+ x2 dA, R= {(x, Y) | -4 s x s 4,0 s y s 1}

A: When the upper and lower limits of definite integrals become same, the result is zero irrespective…

Q: Jse Green's theorem to evaluate line integral | V8 + x3 dx + 9xy dy where C is a triangle with…

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Q: Evaluate the iterated integral. V 16 - 16 – y2 | (x + y) dx dy Submit Apcwor

A: To find the following integralI=∫04∫016-y2x+ydx.dy

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

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Q: 7. Evaluate the line integral f, F· dr where F(x,y, z) = x³i – y²j+ zk and C is the x2 + 2xy + y²…

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Q: Find the differential of w = x ev sin z at the point (x, y, z) = (1, 1, 0). %3D O dy + dz O dy O dx…

A: Obtain the differential of the given function as follows. w=xeysin zdw=ddxxeysin zdx+ddyxeysin…

Q: Evaluate the line integral by two methods: (a) directly and (b) using Green’s Theorem.3. ∮C xy dx +…

A: According to Greens theorem :

Q: Calculate the double integral. JS y dA, 4xye+²y R R= [0, 1] × [0, 8]

A: Given; ∫∫R4xyex2ydA, R=0,1×0,8   As we know the common integral. ∫eax+bdx=1aeax+b+C  And ∬dA=∬dxdy…

Q: The area bounded by y=1-x² and x+y=1 is submerged vertically in the water with the water level in…

A: Solution

Q: Suppose that f(z, y) = /4 – a² – y² at which {(r, y) | 2² + y² < 4}. Then the double integral of…

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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: 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 double integral ∫∫R  y / (1 +xy)2 dA, where R= [0,1]×[0,1].

A: Given integral is  where  R= [0,1]×[0,1].

Q: Evaluate the double integral (x +y)² sin(x – y) dA for R, which is square with the vertices (0, 1),…

A: Consider the given integral and construct a diagram based on the given conditions.

Q: integral Over the é dz Z-2 the Compute %3D Curve C below lm (2) Relz)

A: In the given question we have to find the value of the integral I over the curve C. and the given…

Q: Compute the line integral of the scalar function f(x, y) = /1 + 9xy over the curve y = x' for 0 < x…

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

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Q: Integrate fcx)9,2) =X+y -Z over thep Poth from OLo.0.0) to A (2,1,2) given by, ostsi (C) = r z i…

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Q: The area of the surface obtained by rotating the curve y = 1 I+ X, 1 <xse about the line x = 28 is…

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Q: Evaluate the line integral | F•ds where F = 2xyi + (x – y²)j and C is (a) y = x from (0,0) to (1,1)…

A: For the solution follow the next steps.

Q: 3 Use polar coordinates to evaluate the integral (3) f[/x² +y° dA Ris bounded by the semicircle y =…

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Q: Compute the line integral of the scalar function f(x, y) = over the curve y = x° for 0 <x <1 /1+ 9xy…

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Q: ·볼+1 2x - y dx dy, Use the transformation to evaluate 2x-y where u = v = % and integrating over an…

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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 along the given positively oriented curve. xy2 dx…

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Q: Evaluate the line integral I of the one-form w = (y + z) dx + (x + 2) dy + 10z dz over the line…

A: Given: One-form ω=y+zdx+x+zdy+10zdz and the line segment joining 0, 0, 0 and (3, 1, 0). To find:…

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: Use Green's Theorem to evaluate the line integral. Orient the curve counterclockwise. y10 dx + x6…

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Q: 2) Find the outward flux of F across the region D where F(x, y, z) = (In(x² + y²), –tan'(), zVx² +…

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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: Evaluate the triple integral x dV where G is the solid that lies below the surface z = 15xy and G.…

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Q: g) div(7) for 7(x, y,z) = xyi + yz²j +x²yzk h) curl(ï) for ū(x, y, z) = xzi + xy²j+y²zk

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Q: (b) Let F(x,y) = (e2*y+cos(7x²), xºy^+e>Y³} Use Green's Theorem to set up the line integral F. dr as…

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Q: Evaluate the line integral | F•ds where F = 2xyi +(x – y²)j and C is (a) y=x from (0,0) to (1,1) (b)…

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Q: The area of the surface obtained by rotating the curve y = 1 < x s e about the line x = 24 is O 2n…

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Q: Evaluate the line integral by two methods: (a) directly and (b) using Green's Theorem. C is the…

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Q: Evaluate the iterated integral LL V-6x-x2 ху (x2 + y2)3/2 dydx -3 J0 using polar coordinates.

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Q: Evaluate the triple integral x dV where E is the solid bounded by the paraboloid x = 6y² + 6z² and x…

A: To evaluate the triple integral ∬∫ExdV where E is the solid bounded by the paraboloid…

Q: Use Green's Theorem to evaluate the line integral below. √ (x² − y²) dx + 7xy dy C:r = 1 + cos(8), 0…

A: Given integral is ∫Cx2-y2dx+7xy dy where C: r=1+cosθ,  0≤θ≤2π To Use: Green's Theorem to evaluate…

Q: Evaluate the triple integral. 5xy dV, where E is bounded by the parabolic cylinders y = x² and x =…

A: The point of intersection of the cylinder x=y2 and y=x2 is 0 and 1.

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: Q#03(a): Use Green's theorem to evaluate the line integral f. (x – 3y) dx (4x + y) dy, where C is…

A: Green's theorem

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

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

Q: Evaluate the iterated integral. V 4 - y2 (x + y) dx dy

A: