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Think About It The center of mass of a solid of constant density is shown in the figure. In Exercises 43-46, make a conjecture about how the center of mass
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Calculus
- Kinetic energy of a fluid flow can be computed by ∭V12ρv⋅vdV∭V12ρv⋅vdV, where ρ(x,y,z)ρ(x,y,z) and v(x,y,z)v(x,y,z) are the pointwise fluid density and velocity, respectively. Fluid with uniform density 7π7π flows in the domain bounded by x2+z2=7x2+z2=7 and 0≤y≤670≤y≤67. The velocity of parabolic flow in the given domain is v(x,y,z)=(7−x2−z2)j⃗ v(x,y,z)=(7−x2−z2)j→. Find the kinetic energy of the fluid flow.arrow_forwardA lamina occupies the part of the disk x2 + y2 ≤ 16 in the first quadrant. Find the center of mass of the lamina if the density at any point is proportional to the square of its distance from the origin.arrow_forward*INTEGRAL CALCULUS Show complete solution (with graph) 8. Determine the centroid, C(x̅, y̅, z̅), of the solid formed in the first octant bounded by z + y − 16 = 0 and 2x^2 − 2(16 − y) =0.arrow_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_forwardFine the volume generated by revolving the area bounded by the given curves and lines revolved about the given axis of rotation. Y = 1-x^2 y=0 is revolved about the x=1arrow_forward*INTEGRAL CALCULUS Solve for the volume generated by revolving the given plane area about the given line using the circular ring method. Show complete solution (with graph).8. Within y = x^3, x = 0, y = 8; about x = 2arrow_forward
- Set up the double integral required to find the moment of inertia about the given line of the lamina bounded by the graphs of the equations for the given density. Use a computer algebra system to evaluate the double integral. y = 4 − x2 , y = 0, = k, line : y = 2arrow_forwardvolume generated by rotating the region bounded by y = e^-x^2, y=0, x=0,, and x=1 about the y-axisarrow_forwardCenter of mass of constant-density plates Find the center of mass (centroid) of the following thin, constant-density plates. Sketch the region corresponding to the plate and indicate the location of the center of mass. Use symmetry whenever possible to simplify your work. The half-annulus {(r, θ): 2 ≤ r ≤ 4, 0 ≤ θ ≤ π}arrow_forward
- Multivariable calculus. Let F vector = <x,y,z> and use the Divergence Theorem to calculate the (nonzero) volume of some solid in IR3 by calculating a surface integral. (You can pick the solid).arrow_forwardusing double integration find the volume V(S) of the solid bounded by the surface x=0, z=0, y2=4-x, z=y+2.arrow_forwardCenter of mass of constant-density solids Find the center of mass of the following solid, assuming a constant density of 1. Sketch the region and indicate the location of the centroid. Use symmetry when possible and choose a convenient coordinate system. The tetrahedron in the first octant bounded by z = 1 - x - y andthe coordinate planesarrow_forward
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