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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Chapter 14 Solutions
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_forwardVariable-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_forward
- Hydrodynamic maths obeying Boyle's law, is in motion in a uniform tube of small section, prove that if ? (rho) be the density and v the velocity at a distance x from a fixed point at time t,arrow_forward(a) A triangular lamina with vertices (0,0), (-4,2), (6,2) has the density function δ(x,y) =xy i) Sketch the lamina. ii) Find the mass of the lamina. (b) Find the surface area of the portion of the paraboloid z= 2-x2-y2 above the xy-planearrow_forwardHow to interpret/describe the measure of shape (skewness and kurtosis)? A. Give a template or format B. Example Note: it should be the advanced way. Not just basic structure, i.e., interpreting in in-depth manner. Thanks!arrow_forward
- Fine 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_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_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
- True or False Plus A. In evaluating the moment of a planar lamina, a horizontal strip cannot be used as a representative area. B. The moment of any planar lamina is the product of the mass of the region and its centroid. Choices A. Both A and B are true B. Both A and B are false C. A is true, B is false D. A is false, B is truearrow_forwardVariable-density solids Find the coordinates of the center of mass of the following solid with the given density. The cube in the first octant bounded by the planes x = 2, y = 2,and z = 2, with ρ(x, y, z) = 1 + x + y + zarrow_forwardMultivariable 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_forward
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