Find the flux of F(x,y,z) = sin(xyz) i+ xy j + ze*/5 k across the part of the cylinder y? + 2 z? = 4 that lies above the xy- plane and between the planes x = -1 and x = 1 with upward orientation.

Q: Let S be the portion of the cylinder y = ln x in the first octant whose projection parallel to the…

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Q: A fluid has density 1200 kg/m and flows with velocity = i + yj + zk, where x, y, and z are measured…

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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: Compute the flux of F = 3(x + z)i +j+ 3zk through the surface S given by y = x2 + z2, with 0 0, z 2…

A: flux is determined as shown below

Q: Find the moments and center of mass for the thin plate covering the region between the x-axis and…

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Q: (10) Sketch the solid in the first octant bounded by x + z = 1 and y+ 2z = 2. If 8(x, y, z) = x…

A: The given region lies in the first octant, so x≥0, y≥0 and z≥0. To get the points on the planes,…

Q: Find the moment of inertia about the z-axis of a thin shell of constant density δ cut from the cone…

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Q: Find the limits of integration in cylindrical coordinates for integrating a function ƒ(r, θ, z) over…

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Q: Find the centroid of the solid revolving the area bounded by y=x2 and x=2, and the x-axis about the…

A: Given, Three curves  y=x2 , x=2 and x-axis

Q: Find the moment of inertia of the solid formed by revolving about oy the area bounded by y2 = 2x and…

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Q: What is the solid of revolution obtained when you revolve the region bounded by the curves y=x−1,…

A: We have given curves y=x-1,  y=0 and x=3 And we can plot graph of these curves as We can  see here…

Q: Compute the moments of the semicircle x2 + y2 = R2, y ≥ 0 as line integrals. Verify that the…

A: Given semicircle x2+y2=R2,  y≥0 The figure of this semicircle is shown below. In the above figure,…

Q: Find the moment of inertia of the solid formed by revolving about oy the area bounded by * y2 = 2x…

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Q: Find the moment of inertia about the x-axis of a thin plate of constant density δ covering the…

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Q: Find the x coordinate of the centroid of the solid formed in the first octant bounded above by…

A: Given the solid E is bounded by z=1-y2,x=1,x+y=3 in the first octant. Find the x coordinate of the…

Q: 13. Calculate the flux of F across S.F = (3x, xz, z2) and S is the surface bounded by the paraboloid…

A: Given vector field is  F=3x,xz,z2 we have to find the flux through F where S is the surface bounded…

Q: Find the moment of inertia of the solid formed by revolving about oy the area bounded by y2 : = 2x…

A: The solution are next step

Q: Let S be the closed cylinder bounded by the surface x2 + z2 = 4, and the planes y =1 and y = 4. Use…

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Q: Calculate the scalar line Integral where the function is f(x,y)= 1 / x²+y²+1 and the curve is the…

A: We have given function fx,y=1x2+y2+1 and the curve joining the point 0,0 1,0 1,1 0,1. Let A0,0,…

Q: Find |[F•N dS (flux), where F(x, y,z)= 4x'zi +4y'zj+3z*k, S is the sphere with radius R centered at…

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Q: Consider the velocity field F(x, y, z) = (x+ 5VÝ,y – 10x³, z). Let S be the portion of 2x – y – 5z =…

A: given F→x,y,z=x+5y , y-10x3 , z S: 2x-y-5z=0 bounded by x=4 in first octant (a) find the flux of a…

Q: A solid “trough” of constant density is bounded below by the surface z = 4y2, above by the plane z =…

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Q: Find the volume of the solid bounded from above by the surface z = (64 − 4x^2 − 9y^2)^(1/3) and from…

A: To find - Find the volume of the solid bounded from above by the surface z = 64 - 4x2 - 9y213 and…

Q: Find the equation of the solid generated by revolving the region bounded by y=e-x^2 and the lines…

A: Given equations: y=e-x2  and the lines x=0,x=1,y=0 Graph is, Here we shall use the method of…

Q: Compute the flux of F = (5(z + 2), 5, 52) through the surface S given by y = ² + 2, with 0 0, z 2 0,…

A: We have to find flux.

Q: A curved lamina is in the shape of the surface defined by R(u, v) = (u cos v, u + v, u sin v), where…

A: First we find the elementary surface area ds. ds=| Ru X Rv | dudv

Q: Find the volume of the solid abovez=1 - x² - y², below z= 1, and within the cylinder x² + y? =1, by…

A: Solution is in step 2

Q: Find the volume of the solid in the first octant bounded by 2x + 3y + 4z = 12 and the coordinate…

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Q: Find volume of the solid bounded by z = 0, f(x,y) = x² +y², x² + y2 = 62, 3y = -8x and 17y = 10x…

A: We will find out the required volume.

Q: . Find the centroid of the solid generated by the area bounded by y = sin x and the lines x = 0; x =…

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Q: Find the limits of integration in cylindrical coordinates for integrating a function f(r,0, z) over…

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Q: . Find the centroid of the solid generated by the area bounded by y = sin x and the lines x = 0; x =…

A: Since  the area is rotated about x axis ,the y and z coordinates of centroid are 0, and the centroid…

Q: 7. Find the flux of F(x, y, z) = x î+ y ĵ+ z² k across the sphere of radius 1 centered at the origin…

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Q: 5. Find the flux of F (x,y,z) = y cos(z)i-yx sin(x) j+ xy³k through the outward oriented cube of…

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Q: Find the flux of F= through of the total surface of the closed cone y=(x2+z2)0.5 with 0 < y < 2…

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Q: „ Compute the flux of F = xi + yj + zk through the curved surface of the cylinder x? + y² = 1…

A: Consider the vector field:

Q: 4) Write the integral to calculate the outward flux through F = (-yx2, xy,-z) across the sphere with…

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Q: Use the Divergence Theorem to find the flux of F(x, y, z) = 3x³i + 3x²yj + 24xyk across the surface…

A: Flux of a vector field over a closed surface is equal to the divergence of the field in the volume…

Q: A thin wire is bent into the shape of a line segment from (1, 2) to (−1, 1), and the curve y = x2…

A: Given: The thin wire with shape as line segment from 1,2 to -1,1 and along the curve y=x2 from -1,1…

Q: Find the flux of F=(x, y, z) across the solid that is enclosed by the hemi = 0. Assume an outward or…

A: The Gauss Divergence Theorem states that the volume integral of the divergence of a vector field A…

Q: Write the integral to show the outward flux (using divergence theorem) through F= across the sphere…

A: Given function is, F=yx2,-xy,,z Now, by divergence theorem, ∇→·F→=∂∂xyx2-∂∂yxy+∂∂zz=2xy-x+1 Now, by…

Q: "Determine the center of mass of the solid between  y = x2 + z2 and y = 9 with density δ(x,y,z)=…

A: Kindly go through the solution and let me know in case of any doubt or further clarification in the…

Q: (7) Determine the upward flux of V x F through the part of the elliptical cone C = {(x, y, z)2 =…

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Q: Find the volume of the solid that lies under the plane 4x + 6y - 2z + 19 = 0 and above the rectangle…

A: To find volume set up triple integral ∬∫

Q: Find the line integral of z2 dx + y dy + 4y dz, where C consists of two parts, C, and C,. C, is the…

A: Given line integral ∫Cz2dx+ydy+4ydz, where C consists of two parts C1 and C2. C1 is the intersection…

Q: (2) Let S be the surface of the sphere of radius 4 centered at the origin oriented away from the…

A: We can apply divergence theorem to find this

Q: Find the flux of F = 5zi+ 3y²j+ 2xk through of the total surface of the closed cone y = Vx² + z²,…

A: Flux of vector F through a surface S  can be given by  flux= ∫∫SF.ds We will use Divergence theorem…

Q: This question has several parts that must be completed sequentially. If you skip a part of the…

A: Given y=rcosθ and z=rsinθ

Q: 1) The velocity field of a fluid is V(x, y, z) = (y, –x, 8), the surface S is the part of the sphere…

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