b) Given F(x, Y, z) = (x³ + cosh z) i+ (2y³ – 3x²y)j – (x² + 4y²z) k. Use Gauss's theorem to calculate F.n dS where n is the outward unit normal of o, the surface bounded by the planes, x = 0, z = 0 and TM x+ z = 6, and the parabolic cylinder x = 4 – y².

Q: Find the area of the surface x2 + y2 - z = 0 that is cut from above by the plane z = 4

A: Surface is given by x2+y2=z Area=∫∫R2x2+2y2+1dA       =∫∫R4x2+4y2+1dA Let x=rcosθ and y=rsinθ…

Q: Evaluate the surface integral. ) ر م(ل 2) ds S is the part of the cylinder x² + y2 = 16 that lies…

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Q: Consider the surface S= {(x, y,=)| xy + z =5; x20; y20; z20} in R. Use the Method of Lagrange to…

A: Given  surface S=x,y,x   ; xy+z=5   ;   x≥0 , y≥0  , z≥0 consider the point  x,y,z on the surface S…

Q: 3) Evaluate the surface integral. I| (x²z+ y²z) ds where S is the part of the plane z = 4 + x +y…

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Q: If F= 2y I- 3J +x²k and s the surface parabolic cylinder y= 8x in bounded by the planes OF the 2.…

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Q: Find the sum of maximum and minimum values of the function f(x, y, z) = 6x + 12y + 18z on the elipse…

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Q: 2) Use a change of variables to evaluate ff, zdV; where D is bounded by the paraboloid z = 16 – x² –…

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Q: Calculate the integral I = = • (y² + z²) dx + (z² + x²) dy + (x² + y²) dz where I is the…

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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: Find the sum of maximum and minimum values of the function f(x, y, z) = 9x + 18y + 27z on the elipse…

A: Using Lagrange Multipliers with Two Constraints to find the maximum and minimum values of the given…

Q: Find the surface area of the part of the plane 2x+2y +z=4 that lies inside the cylinder x²+y² = 2.…

A: Given that, the equation of the plane is 2x+2y+z=4 and the equation of the cylinder x2+y2=2. Note…

Q: 18. The volume of the solid that lies between the paraboloid z = 2x² + 2y² and the plane z = 6

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Q: Evaluate the triple integral (x + 2y) dV where E is bounded by the parabolic cylinder y = 7x² and…

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Q: Find the sum of maximum and minimum values of the function f(x, y, z) = 6x + 12y+ 18z on the elipse…

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Q: b) Given F(r, y, z) = (x³ + cosh z) i+ (2y - 3ry)j – (x² + 4y²z) k. Use %3D Gauss's theorem to…

A: 1. The divergence is an operator, which takes in the vector-valued function defining this vector…

Q: 2. Evaluate the triple in tegral where E is the y-2x2+2z20nd the plane y= 8. solid bounded by SI…

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Q: Calculate ff F.n dS where F x'yzi + 2xj + y?k in the solid region S bounded by the parabolic tube z…

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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: 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: Let E be the solid region in the first octant bounded by the coordinate planes, the cylinder…

A: The cylinder x2+y2=5y and the sphere x2+y2+z2=25. Obtain the triple integral: ∫∫∫Ey dV The solid is

Q: Consider the surface S={(x, y, z)| xy+ z= 5; x20; y20; z20} in R3. Use the Method of Lagrange to…

A: The distance between points can be calculated using the distance formula. The distance formula is…

Q: b) Use Gauss theorem to evaluate F.ndS, where F(r, y, 2) = (2x²y² + In z) i – xy³ j – tanh(a² + y³)…

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Q: Verify the Green's theorem for integral x²y dx + xy dy, where C is the boundary described…

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Q: Evaluate fff (3x + 2y)dV, where S is the solid bounded above by the plane z=4, below by plane z=0…

A: ∫∫∫(3x + 2y) dV=∫∫∫(3x + 2y) dxdydz = ∫z = 04∫y = -13∫x =-11(3x + 2y) dxdydz = ∫z = 04∫y = -134y…

Q: (15) Use the divergence theorem to evaluate SSF. ds, where S 3 F(x, y, z) = and 5 is the surface of…

A: To evaluate the given integral using divergence theorem.

Q: 1. Find the surface area of the region S in the plane z = 3x + 2y such that 0 <x < 10 and 0 < y <…

A: We have to find the surface area of the region S in the plane z = 3x + 2y such that 0&lt;=x&lt;=10…

Q: Given a surface 2xy - In (3z- 2) = 10, wherez=f(x, y). Find zx at y =7, z= 1. Leave your answer…

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Q: 2. For the surface z = In(y+ Vx² + y²). (b) Find the linear approximation at the point (3,-4).

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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: 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: Evaluate the triple integral ∭E(x+2y)dV where E is bounded by the parabolic cylinder y=2x2 and 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: b) Given F(x, y, z) = (x³ + cosh z) i+(2y³ – 3x²y) j – (x² + 4y²z) k. Use Gauss's theorem to…

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Q: 17. Find the area of one side of the "winding wall" standing orthogonally on the curve x = y°, 0 < y…

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Q: Let S denote the part of the cone z = Vx2 + y? that lies between the surfaces z = 2 + sin(xy) and z…

A: Given : S denotes the part of the cone z=x2+y2 that liesbetween the surfaces z=2+sinxy and…

Q: Evaluate || F.ndS F = xi + yj+(z² –1)k if and o is the curved surface of the cylinder y = \4-x? from…

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Q: 1. Find the surface area of the cone z = Jx2 + y2 that lies between the plane y = x and the cylinder…

A: Surface area in xy-plane

Q: 4. Evaluate the surface integral ff. F.dS by the divergence theorem where F = (3xy2, yx? -…

A: S is the surface x2+y2≤25, 0≤z≤2 So surface S is a cylinder with radius 5 as shown in below figure.…

Q: Find the sum of maximum and minimum values of the function f(x, y, z) = 8x + 16y + 24z on the elipse…

A: To determine the maxima and minima of the function f(x, y, z) with respect to the function is…

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: 3.Evaluate || F. ds where F= zi+x j +y k and 'S' is the surface of the cylinder x'+ y=1 contained in…

A: Given that ∬S F→·dS→ where  F→=z i^+x j^+y  k^ .......1and  S is the surface of the cylinder x2+y2=1…

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: Evaluate (x² + y) dv G if G is the solid bounded by the parabolic cylinder z = the planes x = 0, x =…

A: Here I am using simple tripple integration to solve this question.

Q: Integrate G(x, y, z) = x2y2 + 4 over the surface cut from the parabolic cylinder y2 + 4z = 16 by the…

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Q: b) Use the Gauss-divergence Theorem to find the surface integral ff, F.n ds, where S is cylinder x²…

A: As stated in question we will use Gauss-divergence Theorem to solve this question.

Q: 1. Use the cylindrical coordinate to evaluate fll. x² + y²dV where G is the solid bounded by…

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Q: 2) Use a change of variables to evaluate , zdV; where Dis bounded by the paraboloid z = 16 – x - 4y?…

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