Example 5: Find fff, Vx2 + z²dV, where E is the region bounded by the paraboloid y = x2 + z2 and the plane y = 4.

Q: The plane a+ y-0 intersects the surface 7+ 2:21 along a curve C. Use Lagrange multipliers to find…

A: We have 2 surfaces intersects to form a curve. We need to find the maximum and minimum of a function…

Q: Let I be the tetrahedron with vertices (2, 0, 0), (0, 5, 0), (0, 0, 3) and the origin. Write f(x, y,…

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Q: (15) Let D be the half-disk x² + y? 0. Compute / | (a + y) dA D

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Q: 6 Consider points P and Q in the x-y plane, with P = (1,0) and Q (0,1). The line integral %3D %3D Q…

A: Introduction :Given, two points P and Q .P= (1,0)Q= (0,1)We have asked for the line integral of

Q: a(x,y) Let G(u, v) = (5u + 2v, 5u+9v) be a map from the uv-plane to the xy-plane. Calculate Jac(G) =…

A: Given that be a map from the -plane to the -plane.The objective is to calculate .

Q: Find the maximum and the minimum values of f(x,y,z) = 5x -8y + 8z on the sphere x² + y² + z² = -

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Q: The centroid of the plane region bounded by the graphs of y = f (x), y = 0, x = 0, and x = 3 is…

A: Given :               The centroid of the plane region bounded by the graph of           y = f(x) ,…

Q: 12. Use the transformation z (u + v), y = (v - 3u) to evaluate the integral ff (4x+8y) dA, where R…

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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) = 11zy(9 – 99x – 9y) over the triangle with vertices (0, 0). (†,0)…

A: Given: The triangle vertices are (0,0),(111,0) and (0,1) f(x,y)=11xy(9-99x-9y)…

Q: 9.) Let E be the tetrahedron with vertices: (0,0,0), (3, 2,0), (0,3,0), (0, 0,2). E is bounded by…

A: The tetrahedron bounded by the planes              from the plane…

Q: If F = xy?i + yz? j + zx2k, and S is the region bounded by the triangle with vertices (1,0,0),…

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Q: Let D be the solid bounded by the ellipsoid x2/a2+y2/b2+z2/c2=1 wher a, b,c>0. Let T be the…

A: Given a solid D bounded by an ellipsoid x2a2+y2b2+z2c2=1, where a, b, c>0. Let T be a…

Q: Q6: Find the residues of a) f(z) = Z² 2Z and (ZH)²(Z²+4)

A: Since you added two questions, find the solution of the first one.

Q: Faith observed that her product sales in a certain town is given by the function of M(u, v) = u²v²…

A: The given function is M(u,v)=u2v2 for (u,v)∈ℝ. The given region is bounded by lines passing through…

Q: Find the envelope of the plane y Z + u X a + u + + b+uc = 1.

A: Given Data: Let us consider the given data, xa+u+yb+u+zc+u=1 To Find: The envelope of the plane.

Q: Eind equatons of the following x²-2y²+2²+ yz =20, [4,6-3) W the tangent plane 6 the normal line to…

A: Solution: f(x,y,z)=x2-2y2+z2+yz-20

Q: Find minimum perpendecular distance between given paraboles. y = x2 + 1 y = x - x2

A: To find the minimum distance between 2 parabolas

Q: 4. Use the method of Lagrange Multipliers to find the point on the plane r+y- 2 = 5 that is closest…

A: Given: The plane equation is x+y-z=5 and the point is -3, 5, 4.

Q: F. dr, where C is the ellipse (x + 3)² + 5(y – 2)² = 16 and F(x, y, z) = x²i+ %3D Pj+ z°k.

A: Given: x+32+5y-22=16 Fx,y,z=x2+y2+z2

Q: to find the use lagrange Multiplier Point p (x1s,Z) Closest to the origion on the Plane 2x +y - 2=5…

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Q: 9.) Let E be the tetrahedron with vertices: (0,0,0), E is bounded by four planes: z = 0, x=0, 2 3…

A: Volume of tetrahedron

Q: The absolute maximum value of ƒ(x, y) = 4xy² — x²y² – xy³ on the triangular region in the xy-plane…

A: Given that, f(x,y) = 4xy2 - x2y2 - xy3 and a triangular  region D with vertices (0,0) ,(6,0) and…