Center of Mass In Exercises 41 and 42, set up the triple
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- Set up the triple integrals required to find the center of mass of the solid tetrahedron whose density is the constant k and has vertices at (0,0,0), (2,0,0), (0,1,0), and (0,0,4). Do Not evaluate the integral, only set it up.arrow_forwardUsing the solid region description, give the integral for a) the mass, b) the center of mass, and c) the moment of inertia about the z axis The solid in the first octant bounded by the coordinate planes and x2 + y2 + z2 = 25 with density function p=kzarrow_forwardSet up a triple integral for the moment of inertia about the z-axis of the solid region Q of density . Do not evaluate the integral. Q = {(x, y, z): −1 ≤ x ≤ 1, −1 ≤ y ≤ 1, 0 ≤ z ≤ 1 − x} = √x2 + y2 + z2arrow_forward
- Setup the iterated double integral that gives the volume of the following solid. Properly identify the height function h = h(x, y) and the region on the xy−plane that defines the solid.arrow_forwardSet-up the double integral to find the mass of the surface S : the part of the plane z = 3 − x − 2y in the first octant, if the mass density at any point on the surface is given by δ(x, y, z) = xz with units of mass per unit area. You do not need to evaluate the double integral.arrow_forwardIntegrationDetermine the volume of the solid below the paraboloid z=x²+3y² and above the region bounded by the planes x=0 ,y=1,y=x and z=0arrow_forward
- Variable-density solids Find the coordinates of the center of mass of the following solid with variable density. R = {(x, y, z): 0 ≤ x ≤ 4, 0 ≤ y ≤ 1, 0 ≤ z ≤ 1};ρ(x, y, z) = 1 + x/2arrow_forwardSet-up the iterated double integral in rectangular coordinates equalto the volume of the solid in the first octant bounded above by the paraboloid z = 1−x2-y2, below by the plane z =3/4, and on the sides by the planes y = x and y = 0.arrow_forwardSetup, but don't evaluate, the integrals which give the volume of the solid formed by revolving the region bounded by y = x2+1, y = x, x = 1, x = 2 about these lines: a) x-axis b) y = -1 c) y = 6 d) y-axis e) x = -3 f) x = 4 g) x = 1arrow_forward
- (a) Find the centroid of the area between the x axis and one arch of y = sin x.(b) Find the volume formed if the area in (a) is rotated about the x axis.(c) Find Ix of a mass of constant density occupying the volume in (b).arrow_forwardUse the description of the solid region to set up the triple integral for (a) the mass, (b) the center of mass, and (c) the moment of inertia about the z-axis. Do not evaluate the integrals. The solid bounded by z = 4 − x2 − y2 and z = 0 with density (x, y, z) = kzarrow_forwardVolumes of solids Use a triple integral to find the volume of thefollowing solid. The solid bounded by x = 0, x = 2, y = z, y = z + 1, z = 0, and z = 4arrow_forward
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