1) Find the outward flux of F across the region D where F= (6x² + 2xy, 2y + x²z, 4x² y³ ) and the region D is cut from the first octant by the cylinder x + y = 4 and the %3D plane z = 3.

Q: Find the centroid of the solid bounded by x² = 4y and y = 5x - x² about the x-axis.

A: We will draw the region and rotate it about x-axis to create a solid of revolution . We will need…

Q: Find the centroid of the region bounded by the parabolas: y x²-4 and y = 0.75x²-3.

A: Given: Parabola equations y=x2-4, y=0·75x2-3To find: The centroid of the region bounded by the given…

Q: Integrate the function f(r,y, z) = Over the entire 3D coordinate system, i.e., all of R'.

A: We have to integrate the function fx, y, z=1x2+y2+z21+x2+y2+z2 over the entire 3D coordinate system,…

Q: Suppose F 2.xz + 3y²) ay + (4yz²) az. (a) Calculate fLL F dS, where S is the shaded surface in…

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Q: 4) Find the centroid of the region that is bounded by the X axis, the Y axis and the line that…

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Q: CD has coordinates C(-8,-2) and D(-4,-12). determine the coordiantes of C'D' if (x,y) -> (1/2x,1/2y)…

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Q: A region R consists of a square bounded by the lines x = -5, x = 5, y = 0, and y = -10 and a half…

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Q: Find the area of the region bounded by the curvesxy= 1,xy= 3,y= 2x, and      y= 3x by using the…

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Q: Find the centroid of the region in the first quadrant bounded by the x-axis, the parabola y² = 5x,…

A: The region in the first quadrant bounded by the given curves is plotted below. See that for the…

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: (b) Integrate (x, y,z)=(y+z)+1 the paraboloid 4x - y-z -0 by the plane x 1. over the surface cut…

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Q: Calculate the line integral f (2x²y + 10x + 6y − 6) dx + (−6xy + 7) dy, - where C' is the rectangle…

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Q: Evaluate the circulation of G = xyi + zj + 4yk around a square of side 4, centered at the origin,…

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Q: Find the centroid of the region between the parabola x + y2 - 2y = 0 and the line x + 2y = 0 in the…

A: Given equation of parabola is, Given equation of line is,

Q: 9. Describe the region in the complex plane where f(z) = cot(z) is holomorphic.

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Q: Find the centroid of the area bounded by 2-4y + 8 = 0, the lines x 0 and x = 4, and the x-axis.

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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 (1.2,…

A: Given: The centroid of the plane region bounded by the graph of y=fx, x=0, y=0, x=3 is 1.2, 1.4To…