A parabolic dome is in the shape of the portion of the elliptic paraboloid z = 1- x2 – y? that lies above the xy-plane. The density at each point (x, y, z) on the parabolic dome is given by 8(x, y, z) = x² /5 – 4z. Use surface integrals to find the mass of the said parabolic dome.

Q: given ∫∫s (x2z) dS and S is the part in z = sqrt(x2+y2), which lies in x2+y2 = 1 calculate the…

A: we have to evaluate ∬Sx2zdS Now, we have z=x2+y2 ⇒z2=x2+y2∂z∂x=xz and…

Q: Find the centre of mass of the region of the shaded area as if revolved around the x-axis, Xcm- 9 y…

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Q: = Find the volume of the ellipsoid x² + y² + 6z² 64.

A: Given,                x2+y2+6z2=64.

Q: Find the surface area of the part of the plane z = 4 + 2x + 3y that lies inside the cylinder z² + y²…

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Q: Consider the lamina in the shape of the triangle whose vertices are (0, 0), (L, 0), and (0,L) for…

A: We have to find the moment of inertia of the lamina whose shape is a triangle with vertices  and…

Q: A lamina occupies the part of the disk x² + y² ≤ 16 in the first quadrant. Find its center of mass…

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Q: Find the volume of the solid enclosed by the paraboloids z = 1 (x2 + y²) and z 32 – 1 (x2 + y²).

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Q: Compute the surface area of the domed structure D:= {(x, y, 2) ; x² + y² + (z – 1)² = 2 ; z > 0}.

A: x2 + y2 + (z-1)2 = 2  ,  x2 + y2 + (z-1)2 = 2  ,  z ≥ 0 This equation is for sphere of radius  = 2…

Q: Find the area of the surface. T'part of the paraboloid x = y² + z² that lies inside the cylinder y²…

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Q: Use cylindrical coordinates to determine the volume of the solid that is enclosed by the cone z =…

A: Volume by cylindrical coordinates

Q: Use cylindrical coordinates to find the volume of the solid that lies within the sphere x² + y² + z²…

A: Solid that lies within the sphere above the xy plane and outside the cone .To Find:Using the…

Q: Find the center of mass and the moment of inertia about the y-axis of a thin shell of constant…

A: According to the question,

Q: 3. Find the average distance from the origin to a point inside the part of the cone x² + y² = z²2…

A: For the average distance from the origin to a point inside the given cone, between the planes The…