13. Calculate the flux of F across S.F = (3x, xz, z2) and S is the surface bounded by the paraboloid z = 4 – x2 – y? and the ry-plane. Sketch the solid and the region of integra- tion.

Q: Calculate F· dS where F = (3xy', ze", 2°) and S is the surface of the solid bounded by the cylinder…

A: We need to calculate ∬SF→.dS which is the flux. Now, we can use the divergence theorem to find…

Q: Find [[fw ryz dV if W is the solid region in R³ below the surface z = x² + y² and above the square 0…

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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: 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) Integrate g(x,y,z) = Inx+y over the surface of the region cut from the plane 4x+3y+2z=1 by the…

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Q: 2. Evaluate the surface integral f,yz ds ; o is the part of the plane x + y + z = 2 that lies in the…

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Q: . Evaluate (x² + y) dv G if G is the solid bounded by the parabolic cylinder z = y², xy-plane and…

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Q: Evaluate JS yz + 4ryds where S is the surface of the solid bounded by 4x+2y+z=84, z=0, y=0 and x=0.…

A: Use the divergence theorem to find the surface integral,…

Q: If E is the solid shape bounded by x 2 0,y 20 ,z 2 0 and the plane 2x + 3y +z = 6 evaluate (x + y)…

A: The given question is solved below

Q: Use the Divergence Theorem to calculate the surface integralI F. dS; that is, calculate the flux of…

A: To use the divergence theorem to calculate the flux of F across the surface S. F(x, y, z) = 3xy2i +…

Q: 4. Find the volume V of the solid S that is bounded by the elliptic paraboloid defined by 2x2 + y2…

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Q: a)Calculate the integral [D zdV on the D region bounded by the 6 = x2 + y2 paraboloid from the…

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Q: Verify Green's theorem in the plane for f (32 - Sy*)dz + (4y – 6zy)dy, where C is the boundary of…

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Q: 5. Evaluate the surface integral S_(x+ y) ds ; o is the part of the plane 2x + y +z = 3 that lies in…

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Q: Evaluate the surface integral. z + x2y x²y) ds S is the part of the cylinder y² + z2 = 9 that lies…

A: We have to find the surface integral from the given condition

Q: Evaluate the surface integral. z2 S is the part of the paraboloid x = y2 + z² given by 0 sx 5 4 1 V5…

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Q: 4. Let S be the portion of the paraboloid z = 4 – x² – y? in the first octant. (a) Evaluate the…

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Q: Use Stokes' theorem to calculate JJ (A) nds by A = 3yi-xzi+yz²k where S is the surface of the…

A: The given vector field is: A=3yi-xzj +yz2k The surface S is the paraboloid 2z=x2+y2 bounded by z=2.…

Q: (1) Use the divergence theorem to evaluate F-S. where F (z, y, 2) = zzi – j+4:k and S is the surface…

A: we use divergence theorem for finding the surface integral.

Q: Evaluate the surface integral. z2 dS, S is the part of the paraboloid  x = y2 + z2  given by  0 ≤…

A: given the surface integral  z2 dS  where S is the part of the paraboloid x= y2+z2   given by 0≤x≤1…

Q: Calculate F. dS where F = (e" sin y, e² cos y, yz²) S is the surface of the box bounded by the…

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Q: Evaluate SſF-dš where F = yi +2x j +(z-8)k and S is the surface of the solid bounded by 4x + 2y +z =…

A: By using the divergence theorem we solve the above question

Q: Calculate F. dS where F = (e* sin y, e² cos y, yz²) S is the surface of the box bounded by the…

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Q: 6. Below is the solid bounded by the surfaces S1:z° = 16 – 2y S2: x = 4-y + z, S3: x = 1, in the…

A: Given a solid described by,  s1:z2=16-2ys2:x=4-y+zs3:x=1.

Q: 8. Use Green's Theorem to calculate f, x?y³dx + (xy - y?)dy where C is the boundary of the region…

A: The given integral is ∫Cx2y3dx+xy-y2dy, where C is the boundary of the region lying between the…

Q: #3. Evaluate the surface integral xz ds, where S is the part of the plane 2x +2y +z = 4 that lies in…

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Q: Evaluate the surface integral. (z + x2y) ds S is the part of the cylinder y2 + z2 = 16 that lies…

A: Given ∫∫S(z+x2y)dS where S is the part of the cylinder y2+z2=16 that lies between the planes x=0 and…

Q: Calculate F- dS where F = (e" sin y, e* cos y, yz²) S is the surface of the box bounded by the…

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Q: If E is the solid shape bounded by x 20,y 20,z 20 and the plane 6x + 4y + 3z = 12 evaluate %3D (y +…

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Q: 3. Find the volume under the paraboloid 3z = 2.12 + 2y2, above the triangle enclosed by the lines y…

A: First we have to draw graph and find limit according to which we have taken strip along y . Find…

Q: a) Integrate g(x,y,z) = Inx²+y over the surface of the region cut from the plane 4x+3y+2z=1 by the…

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Q: Evaluate the triple integral (* + 1y) dV where E is bounded by the parabolic cylinder y = 32² and…

A: We have to find the triple integral  ∫∫∫E(x+y)dVwhere E is bounded by the parabolic cylinder y=3x2…

Q: Let S be the negatively oriented surface defined by R(u, v) = (u², -uv, v²), where (u, v) € [0.2] ×…

A: Given Ru,v=u2,-uv,v2 and Fx,y,z=y,-xz,y Now, Ruu,v=2u,-v,0Rvu,v=0,-u,2v So,…

Q: 10/ Compute e Fids, where S is the surface z =N16-x²-y² oriented up ward and F = <y,-x, zZ %3D 工=?

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Q: Use Green's Theorem to calculate S,x²y³dx + (xy – y²)dy where C is the boundary of the region lying…

A: Greens Theorem:

Q: Evaluate x dV , where E is the E region in the first octant bounded by the sphere x² + y² + z² = 1…

A: given region is in the first octant. using spherical coordinates x=ρsinϕcosθy=ρsinϕsinθz=ρcosϕ and…

Q: Calculate F- dS where F = (3ry², re², z³) S is the surface of the solid bounded by the cylinder y? +…

A: The general form of a vector-valued function in three dimensions is given by F¯=F1,F2,F3, where F1,…

Q: (a) Compute the surface integral || a xy dS, where S is the quarter cylinder x2 + y? = 1; x, y > 0…

A: Given that The surface integral is ∬Sxy dS, where S is the quarter cylinder x2+y2=1; x,y≥0 between…

Q: please help me as soon as possbile

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Q: A solid G is bounded by the paraboloid z = 4 – r - y², the cylinder 2+ y = 4 and the plane z = 5.…

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Q: E) Sketch the solid bounded by the cylinder ** +y = 4 and the planes y+z = 4 and z = 0

A: Here, we need to sketch the solid bounded by the cylinder x2+y2=4 and the planes y+z=4 and z=0.

Q: Find the flux of F(x, y, 2) = (r, y, 2) across the surface o which is the surface of the solid…

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Q: E is the region behind the surface y = 4 – x² that is in front of the region in the xy-plane bounded…

A: Given integral: ∫∬E18x dV Given that E is the region behind the surface y=4-x2 that is in front of…

Q: c, y, z) = x²zi + xz'j+ y ln(x + 1) k, s the surface of the solid bounded by the 2z = 4, y = 3, x =…

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Q: Verify Stokes' Theorem for F = xyî + yzj + xzk and where s is the surface of the paraboloid z = x² +…

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Q: Let S be the piece of the surface z = 9-r that lies inside the cylinder r2 +y² = 9, and let F = (x,…

A: Given, S be the piece of the surface z=9-x2 that lies inside the cylinder x2+y2=9, and let F=x,y2,z.…

Q: Let the surface Σ be the solid bounded by the paraboloid z “ 1 ´ x2 ´ y2 and the xy´plane.…

A: Use Gauss divergence theorem to compute the outward flux.

Q: Use Stokes' theorem to calculate (V × A) · ndS with A = 3yi – xzj+ yz²k, where S is the surface of…

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Q: Verify Green's theorem in the plane for (3x² -8y²)dx+(4y-6xy)dy, where C is the closed curve of the…

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