17. Integrate G(x, y, z) = xyz over the triangular surface with vertices (1, 0, 0), (0, 2, 0), and (0, 1, 1). (0, 1, 1) 1 (0, 2, 0) x* (1, 0, 0)

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Q: 3. Compute the scalar line integral 3 y – xz ds where C is the line seg- ment from (0, 1, 1) to…

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Q: 6. The solid bounded by x = y², and x = 8 – 2z2 – y?.

A: To find the triple integral over the given solid bounded by x=y2,x=8-2z2-y2.

Q: Integrate G(x, y, z) = xyz over the surface of the cube cut from the first octant by the planes x =…

A: Here, the surface is in the first octant. Therefore, the integration is

Q: EXAMPLE 2 to evaluate the Use the change of variables x = u² - v², y = auv integral SSR y dA, where…

A: The objective is to evaluate the integral using change of variable. x=u2-v2, y=2uvy2=4-4x, y2=4+4x,…

Q: 1. Under the paraboloid = 3z2 + y? and above the region bounded by y = x and r= y² – y

A: Find point of intersection of:  y=xx=y2-y y=y2-yy2-2y=0y=0,2

Q: d. .2 The solid bounded by the cylinder x+y = 9 and the planes z = 0 and z = 3- x. %3D

A: The solution of the problem is given in the next steps.

Q: 39.) The solid bounded by the cylinder x+y² = 9 and the planes z= 0 and z = 3 - x. 40

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Q: Show the triangular region bounded by the lines x = 0, y = 0 and x + y = 1 is invariant for the…

A: Given: It is given that, the triangular region is x=0, y=0 and x+y=1. To prove: We have to prove…

Q: The tangent plane and the normal line of the surface y = x² + z – 6 at the point (2, -1, 1) have the…

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Q: F - dr where F(x, y, z) = (4.25yz cos(xyz), 4.25xz cos(xyz), 4.25zy cos(ryz)) Evaluate п 1 and Cis…

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Q: 1) Find the outward flux of F across the region D where F= (6x² + 2xy, 2y + x²z, 4x² y³ ) and the…

A: The vector field is,F¯=6x2+2xy, 2y+x2z, 4x2y3The region D is cut from the first octant by the…

Q: Find the maximum value of f(x, y) = 9xy(5 – 25x – 5y) over the triangle with vertices (0,0), (, 0)…

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Q: (2) Find the integral (0,2,0) and (0,0,1). where D is the tetrahedron with vertices (0,0,0),…

A: Given : ∬∫x+2y+z dV             where D is  tetrahedron with vertices 0 ,0 ,0  , 1 , 0 , 0  , 0, 2 ,…

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: Find parametric equations for the normal line to the surface x² + 9xyz + y² = 11z² at the point (1,…

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Q: Find and classify all the critical points of the surface with its equation z = 32°y – y° + 3z – 3y²…

A: Given: z = 3x2y - y3 + 3x2 -3y2 + 2

Q: 1. The solid bounded by the cylinder y = 9 – x² and the paraboloid y = 2x? + 3z².

A: In this question, we find the volume bounded by given surfaces. The surfaces: y=9-x2andy=2x2+3y2 The…

Q: 43. Find f, 1dS, where s is the portion of the surface deter- mined by z = x2 – V3y that lies above…

A: Find ∫s1 dS where S is the portion of the surface determined by z=x2-3y that lies above the region…

Q: The solid bounded by the surfaces S1: −2 (z - 2) = y2 S2: x = 5 - 2z S3: x = 0 S4: y = 0 S5: z = 0…

A: Given that, A solid is bound by the surfaces, S1: -2(z-2)=y2S2: x=5-2zS3: x=0S4: y=0S5: z=0

Q: Find parametric equations for the normal line to the surface x2 + 9xyz + y² = 11z2 at the point (1,…

A: Given:- x2+9xyz+y2 =11z2  To find:- Equation of normal line to the given surface at point (1,1,1)

Q: (a) Between 1 and the disk x²+ (1+x² +y*) (b) Between % = 1 and the entire xy-plane. (1+x* + y?)*

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Q: 1.13 Evaluate the line integral $ x*dx + xydx where C is the triangle with vertices at (0, 0), (0,…

A: The given function is x4dx+xydy By green's theorem, the integral becomes, ∫06∫06∂M∂y-∂N∂xdxdy where…

Q: Find and classify all the critical points of the surface with its equation z = 3x"y + y* – 3x2 – 3y?…

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Q: 9. Evaluate the line integral / (2x – 3y) ds; where C is the line segment from (-1,0) to (0, 1)…

A: 9.

Q: The solid bounded by y = 4, z = x², z = 2y, and z = 4.

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Q: 1_2 Can you please help me with this example in a step by step guide. Thank you very much!!

A: To calculate the coordinates (x,y,z) and also define the absolute extremum for the provided surface…

Q: 4. What is the maximum value of f(x, y, z) = xyz for points (x, y, z) lying on the ellipsoid x² + 4…

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Q: |Evaluate the line integral: f.(r² + y) dx + (4x – cos(y)) dy where C is the boundary of the square…

A: According to Greens theorem, the line integral ∮CPdx+Qdy=∬DQx-PydA. BY the extended version of…

Q: termine the solid bounded by the surfaces 9x2+4y2=30 and the plane 9x+4y−6z=0 in the first octan

A: The given solid is bounded by the surface 9x2+4y2=30 and the plane 9x+4y−6z=0 in the first octant.

Q: The equation of the tangent plane to the surface x² 8 at the point (2,-1, 3) is Select one: O x-2…

A: In this question, we want to find the equation of the tangent plane to the given surface.

Q: Find the center of mass of a thin triangular plate bounded by the y-axis and the lines y = x and y =…

A: Given problem is :

Q: Evaluate the solid bounded by 2x+z=2 and (x-1)2+y2=z.

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Q: Show that 1 y - X 1. Z-y X-Z = 1. %3D alx-y) alz - x)

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Q: The solid bounded by z = x, z = 2 – x², z + y = 3, and y = 0

A: Volume of a solid is obtained using triple integration by the formulae =∫∫∫dx dy dz The solid is…

Q: 16. The solid within the spherex² + y² + z² = 9, outside the cone z = Vx2 + y², and above the…

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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: 3.4.6 Use Green's theorem in a plane to evaluate of 10x9²³- [(xy²-y) dx + (x + y²) dy] C as a double…

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Q: 2. The area bounded by y = -(x - 1)2 + 4, and the lines x =1, y = 2, and y = 4 rotated about the %3D…

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Q: Evaluate the line integralf, (xy + z)ds along the line segment C from (1,2,3) to (-1,1,5).

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Q: 5) Find the region in the xy-plane for which (sin x – y)y' = vx- y satisfies the nonlinear existence…

A: A differential equation  y' =f(x,y) will have at least one solution in the region where f(x,y) is…

Q: (6) Evaluate the line integral / r²ydx + xy°dy, where C is the square with vertices (0,0), (1,0),…

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Q: find the points of the surface x^2-y^2+z^2=1 which corresponds to the extreme value of…

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Q: The minimum value of f(x,y,z) = x^2 -2y + 2z^2 on the sphere x^2 + y^2 + z^2 =1 is ....?

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Q: #1. Evaluate (2x – 5y) dA, R where R is the parallelogram with vertices (0,0), (2,3), (–1,4) and…

A: The given region R is a parallelogram with vertices as shown in below graph. We have to use change…

Q: 2 : Use Lagrange multipliers method to show that the maximum value of f(x, y, z) = x²y²z² on the…

A: In lagrange method, the Critical points are the points that cancel the first order partial…

Q: 8. Evaluate the line integral fx'dx + xe 'dy where C is the closed path along the triangle in the…

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Q: 2. Find the volune og the solid gonned by revohing abaut thie lere y y=4x+1, y: -1 bounded by 4x +3,…

A: The volume for solid of revolution generated by revolving the area bounded by curves y=f(x) and…

Q: 10. The solid bounded by y = 4, y – x = 1, z = x², and z = = 4.

A: