Consider the solid λ bounded by the xz-plane, the yz-plane, the plane z = −1, the plane x+y = 3, and the cylinder z = 8 − x2 as shown in the picture attached.  Let C be the volume of the solid λ, and suppose that the density at each point (x, y, z) in λ is given by δ(x, y, z) = 3. Express the x-coordinate of the center of mass of λ in terms of C, W, and constants.

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: Let a and h be positive real numbers. Let E be the surface in R³ given by 4a.x = y? + z2. (a) Find…

A: Given 4ax=y2+z2

Q: A fluid has density 900 kg/m³ and flows with velocity v = zi+yj + zk, where x, y, and z are measured…

A: Given: Density ρx,y,z=900 kg/m3  , v→=xi^+yj^+zk^. To find: The rate of flow outward through the…

Q: 2. A solid is in the first octant lies inside the hemisphere z = 4 - x² - y², outside the cone z =…

A:

Q: Find the centroid of the region W lying above the sphere x² + y² + z² = 30 and below the paraboloid…

A: Given the sphere x2+y2+z2=30 and below the paraboloid z=18-x2-y2. And the density of δx, y, z=1.

Q: The base of a solid is the region in the xy-plane bounded by the curves z=-y² + 8y + 113 and x = y²…

A: Given that : Every cross-section of this solid perpendicular to the y-axis (and to the xy-plane) is…

Q: Find the volume of the solid under z = 6 - 5x and above the plane bounded by y = 2x, y = 2, x = 0,…

A: Given Curves are                                                      z=6-5xy=2xy=2, x=0 volume of…

Q: Find the volume bounded by the paraboloid z = 1+3x² +3y² and the plane z = 10 in the first octant.

A:

Q: Find the volume of the solid that lies under the plane 3x + 2y + z = 12 and above the rectangle. R =…

A: The given plane is:3x+2y+z=12The region in x-y plane is:R= (x, y) | 0≤x≤1,  -4≤y≤4

Q: /|F . d5 where Calculate F = (3zy°, ze", 2ª) Sis the surface of the solid bounded by the cylinder y?…

A: