(a) Given F(x, y, z) = –xi– y j+x²k and o is the surface of the solid bounded by the Vx² + y² and the plane z = 1 and z = 2. By using the Gauss's theorem, cone z = %3D obtain F.ndS.

Q: 2. Evaluate the triple integral //y o 2 = y, the xy-plane, and the parabolic cylinder y = 1- r2. y…

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Q: SSL. xy dV, where T is the first-octant solid bounded by the coordinate planes and the upper half of…

A: Here we have to find  ∫ ∫ ∫T xy dV , where T is the first-octant solid bounded by the coordinate…

Q: find the outward flux of F across the boundary of D. F =-2x i - 3y j + z k D: The upper region cut…

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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: Find fff y av, y dV, where E is the solid bounded by the parabolic cylinder z = ² and the planes y =…

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Q: The surface area of the part of z = 2 + V4 – x² – y² that lies inside the paraboloid z x² + y² is

A: This is a problem of vector calculus.

Q: a) Let G be the wedge in the first octant that is cut from the cylindrical solid y² + z? < 4 by the…

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Q: Consider the solid bounded by surfaces S1, S2, S3, S4, S5, and S6 with vertices A, B, C, D, E and O,…

A: We need to find the length of the curve BD. The curve BD is found by the intersection of the…

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

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Q: Find the sum of maximum and minimum values of the function f(x, y, z) = 4x + 8y + 12z on the elipse…

A: Lagrange's multiplier method uses the system of the equations obtained from the partial derivatives.…

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

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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) ||| Va² +g³° dv E where E is the solid that lies within the cylinder x²+y° =1, below the plane…

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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: Find the centroid (δ = 1) of the solid enclosed by the cylinder x2 + y2 = 4, bounded above by the…

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Q: Calculate F- dS where F = (3zy', ze", z) S is the surface of the solid bounded by the cylinder y? +…

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Q: Find the sum of maximum and minimum values of the function f(x, y, z) = 4x + 8y + 12z on the elipse…

A: Using Lagrange Multipliers with Two Constraints

Q: a) Find the points at which the twisted cubic F(t) = ti + t} +tk intersects the plane 4.x +2y+z = 24…

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Q: É = ( 2x³ + y³, y³ + 2³, 3y²z), S is the surface of the solid bounded by the paraboloid z = 1– x2 –…

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Q: - y² + 4y + 49 and The base of a solid is the region in the zy-plane bounded by the curves a : x =…

A: Given , base of a solid = region in the xy-plane bounded by  the curves…

Q: • SSS ²³² x² + y² = 2, and the xy plane. Evaluate e² dV where E is enclosed by the paraboloid z = 6…

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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: 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: Find F.dS if F (x, y, z) = 2xi+2yj+2zk and S is t he part of the paraboloid z = 8-x²–-y? lying above…

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Q: The base of a solid is the region in the xy-plane between the the lines y = x, y = 4x, x = 1 and x =…

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Q: Find /// y dV, where E is the solid bounded by the parabolic cylinder z = 2 and the planes y = 0 and…

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Q: Find vdv y dV, where E is the solid bounded by the parabolic cylinder z = x and the planes y = 0 and…

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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: Consider the PDE y u + Uyy + u, - Uy = 2 +y? In what region of the xy-plane is this PDE elliptic? xy…

A: The objective is to find in what region of the xy-plane the pde is elliptic.

Q: 11. Integrate G(x, y, z)=x√y² + 4 over the surface cut from the parabolic cylinder y² + 4z = 16 by…

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Q: E is the solid in R3 bounded by the cone surface z = square root (x2+ y2 ) and the plane z = 3, and…

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Q: /| y dV, where E is the solid bounded by the parabolic cylinder z = 1° and the planes y = 0 Find and…

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Q: É = ( 2x3 + y°, y3 + z³, 3y²z), S is the surface of the solid bounded by the paraboloid z = 1– x2 –…

A: We will find out the required flux.

Q: Evaluate the surface integral (4x² + 3y)dS where S is the part of the plane 2.x + 5y + z = 48 inside…

A: Surface integral is a general form of multiple integrals over a surface. Sometimes it is same as…

Q: Find the orthogonal trajectories of the conicoid (x + y)z = 1 of the conics in which it is cut by…

A: Given:The equation of conicoid is (x+y)z=1The system of the plane is x-y+z=K, where k is…

Q: Integrate g(x, y, z) = x4y(y2 + z2 ) over the portion of the cylinder y2 + z2 = 25 that lies in the…

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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: Q– Fin the surface area of the portion of the paraboloid z= 6-2x - y that is above the x-y plane…

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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: Find the sum of maximum and minimum values of the function f(x, y, z) = 4x + 8y+ 12z on the elipse…

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Q: Consider the solid region E bounded by the paraboloids z = x^2 + y^2 and z = 36 − 3x^2 − 3y^2

A: We are given that the solid is bounded by the paraboloids z=x2+y2 and z=36-3x2-3y2. For (a), We need…

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: 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: The paraboloid 1. volume 2 %3D 1 2. volume = T is shown in 3. volume = 1 4. volume 2 7 %3D 5. volume…

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Q: //| y dV, where E is the solid bounded by the parabolic cylinder z = x² and the planes y = 0 E Find…

A: we have to find volume

Q: 10. Use cylindrical coordinates to evaluate SS S(x+y+z) dV where E is the solid enclosed by the…

A: The region

Q: Find SS F dS where F(x, y, z) = z i+(3x – 1)j+ yk and S is the portion of the plane x + y + z = 1…

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