Let G(x, y, z)= (1+9xy)2 and consider a surface S given by the parametrization r(u, v)= [u, u, v], 0sus1, -1

Q: 1. The solid bounded by y = 1, y = x², z = 0 and z = y AS A Z-SIMPLE SOLID.

A: Givne that  Solid region y=1,y=x^2,z=0 and z=y  To find the triple integral

Q: Let S denote the portion of the surface z=x + 4 above the triangle on the xy plane with vertices…

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Q: Let R = {(x, y) E R² | In y <x <1n1< y<e} and let S be the surface defined by the vector function…

A: As per the question we are given a surface S defined by the vector function R(u,v) And we have to…

Q: Evaluate the integral for y dx + z*dy + x²dz, where C is the triangular closed path joining the…

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Q: 5. (a) Evaluate the surface integral /3² dS where o is part of the cone z = VT + y² that lies…

A: As you post multiple questions According to the guideline I only solve the first problem please post…

Q: 4. Let S be the open surface, x² + y² + z² = 4; z > 0. Evaluate the following surface integral,…

A: Given that S is the open surface x2+y2+z2=4, z≥0 To evaluate ∬SF·ndS Here, F=x,0,0 Let us…

Q: 6. Let S be the sphere 2 + y² + z² = 1 and F(x, y, z) = (x + e*+y, y+ In(sin(z)), z). Calculate the…

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Q: (a) Evaluate ([ Á.ds where A=(x+y²)î-2xj+2yzk and s is the surface of the plane in the first octant…

A: For the detailed solution of the problem follow the next steps.

Q: 6. Compute the area of the surface for 0 <u 1, 0 < v < using the first fundamental form..

A: Given xu,v=u2-cosv,u+sint,cosv

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: 11. Find the F in the surface S integration ffrnd where n is the normal unit vector to the S F=xi+zk…

A: Given F=xi+zkS : 2x+y-2z=6  where x≥0, y≥0, z≤0

Q: 5. Let S be the negatively oriented surface defined by R(u, v) = (u², v², uv), where (u, v) = [0, 1]…

A: Given That : R(u,v) →=u2,v2,uvF(x.y,z)→=z,z,xy Find : Double integral which is equal to the flux of…

Q: Let G(x, y, z)= (1+9xy)² and consider a surface S given by the parametrization r(u, v)=[u, u³, v], 0…

A: In this question, we find the surface area integral for a given parameterized surface in terms of u…

Q: 4. Use Green's Theorem to evaluate the line integral | 2xydx+ xy'dy, where C is the triangle with…

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Q: 7. Use Stokes's Theorem to write S.(2y² + 6x*)dx + (2x² + 7y³)dy+ (2z° – yz)dz as a surface integral…

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Q: Use Green's Theorem to evaluate the line integral: f. In(1 + y) dr – #, dy where C is triangle with…

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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: 5- Evaluate the surface integral ffs G(r)dA, where G = arctan(y/x) and S is the surface z = x2 +…

A: To evaluate ∬SGrdA where G=arctanyx and S is the surface z=x2+y2, 0≤z≤9, 0≤x, 0≤y Let us…

Q: 5. Use our Early Parameterization step by step including sketch of the surface to evaluate the…

A: Given that The equation of cylinder in the xz-plane as x2+z2=1 Also, the given surface integral lies…

Q: 2.8 Using the divergence theorem, show that the volume, V, enclosed by the surface I can be obtained…

A: As per the Gauss Divergence Theorem, the surface integral of the normal component of G vector…

Q: 5. Let S be the negatively oriented surface defined by R(u, v) = (u², v², uv), where (u, v) [0, 1] x…

A: This is the problem from Vector Calculus.

Q: the value of the double integral ∬ R (y^2 + x^2) dA, where R is a circular disk defined by x^2 + y^2…

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Q: Let R {(x, y) E R² | In y <x <1n1<yse} and let S be the surface defined by the vector function R(u,…

A: First we find the limit of integration for u and v

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 in order to compute the line integral (2sin z + 3y²) dx + (sin(4®) + 2x³) dy…

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Q: 2- Use Gauss's divergence theorem to evaluate the surface integral || (ry dydz + 2y°dædz + y°…

A: To evaluate the surface integral: ∬Sxy2dydz+2y3dxdz+y2zdxdy

Q: Set up the surface integral for upward Flux of the upper hemisphere of radius = 5 (z20) such…

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Q: 4) Evaluate the surface integral [[F•dS where F(x,y,z)= (yz,-xz,z)and S is the piece of the…

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Q: 4. Use Green's Theorem to evaluate the line integral | 2xydx+ xy*dy, where C is the triangle with…

A: The given triangle C with the vertices (0,0),(1,0) and (1,2) is as as shown below.

Q: Use Stokes's Theorem to evaluate F(x, y, z) = yzi + (2 − 6y)j + (x² + y²)k, x² + y² ≤ 49 S: the…

A: Let's solve part 2.

Q: 5- Evaluate the surface integral SS,G(r)dA, where G = arctan(y/x) and S is the surface z = a² + y²,0…

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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: a) Evaluate the surface integral 2x²y dS over the surface y² + z² = 1 between x = -1 and x = 5.

A: Evaluate the surface integral ∫∫δ 2x2y dS over the surface y2+z2=1 between x=-1 and x=5. If ru, v is…

Q: 18. z = 4 – x2 – y², 0<z< 3; f (x, y, z) = x² / (4 – 2) - -

A: Use polar coordinates to solve the integral

Q: Given F(x, y,z) = x*y² i+ y°zj+xz°k_ and S is the surface of the sphere x* +y° +z = 4. Find the…

A: I am going to solve the gven problem by using some simple calculus to get the rquired result.

Q: Set up and evaluate the surface integral ||(4x+z+6)dS over the surface S ř (u,v)=(v, 4cos u, 4sin u)…

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Q: An iterated double integral for evaluating the surface integral S f,(xy+yz) dS where E is the part…

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Q: Verify Stokes' Theorum for F(x,y,z) = (y,-x,yz) if S is the paraboloid z = x2 + y2 with x2 + y2 = 1,…

A: Given  Fx,y,z=y,-x,yz Then, F→=yi^-xj^+yzk^

Q: 11. Consider F(x, y, z) = (-y, x, e²) and the surface defined by S = {(x, y, z) : z =1– 2² – y², x²…

A: Given F=-y,x,ez and S=x,y,z| z=1-x2-y2,x2+y2≤1. Verify the stokes theorem ∫CF·dr→=∬Scurl F·n→ dS.

Q: 8. Use the Divergence Theorem to calculate the surface integral for F(x, y, z) = (x³ +y³)i+ (y³ +…

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Q: 4. Consider a soup can of height 2 and radius 3, whose base (bottom) lies in the r- y plane and is…

A: Given- Consider a soup can of height 2 and radius 3, whose base (bottom) lies in the x-y plane and…

Q: Use Stockes' theorem to calculate the line integral fe F.dr where F (2.52) = (szy + e)i + (2 – Iny)3…

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Q: Use Stokes' theorem to evaluate line integral (z dx + x dy + y dz), where C is a triangle with…

A: Consider the provided question, We have to evaluate the given integral by using the stokes theorem,…

Q: Evaluate the surface integral. ds Is 3/2 + y3/2), o s xs 1, 0sys 3 0sx < 1, 0 s ys 3 S is the…

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Q: Let C be the intersection curve of the paraboloid z = x2 − y2/2 with the cylinder x2 + y2 = 1,…

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Q: Let G(x, y, z)= (1+9xy)² and consider a surface S given by the parametrizatio Osus2,0svs2. Then the…

A: Here we have,    and surface S by the parametrization We have to find,

Q: (2) Let S be the surface of the sphere of radius 4 centered at the origin oriented away from the…

A: We can apply divergence theorem to find this

Q: This question has several parts that must be completed sequentially. If you skip a part of the…

A: Given y=rcosθ and z=rsinθ

Q: 3. Evaluate the triple integral || ryz dV; G is the solid in the first octant that is bounded by the…

A: To evaluate the given integral ∫∫∫Gxyz dV. The limits are given as  z=0 to z=2-x2 and y=0 to y=x and…

Q: 3- Use Stokes theorem to compute the line integral SF· dr, where F = [e-3y, e², e-2*] and C is the…

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