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Find the specified quantities for the solids described:

The moment of inertia about the z-axis of the region from Exercise 53, assuming that the density at every point is inversely proportional to the point’s distance from the z-axis.

Short Answer

Expert verified

The moment of Inertia isIz=16kπ3units

Step by step solution

01

Given Information

The density at every point is inversely proportional to the point’s distance from the

z-axis. The equations arex2+y2-z2=1andx2+y2=5.

02

Evaluation of limits

The relation between cylindrical and rectangular coordinates are

r=x2+y2,tanθ=yx,z=z

and

x=rcosθ,y=rsinθ,z=z

The equation of sphere x2+y2-z2=1in terms of cylindrical coordinates gives

r2-z2=1

The equation of cylinder x2+y2=5in terms of cylindrical coordinates yields

r2=5

Rectangular limits are

x2+y2-z2=1z=r2-1(z=0is equation of xyplane)

x2+y2=5r=5

and z=0r=1

Cylindrical limits are0zr2-1,1r5,0θ2π

03

Calculation of Inertia

Moment of inertia is given by Iz=Ex2+y2ρ(x,y,z)dxdydz

Since the density at every point is inversely proportional to the point’s distance from the zaxis.

ρ(x,y,z)=kx2+y2

Putting limits

role="math" localid="1652386511310" Iz=θ=02πr=15z=0r2-1krdzdrdθ

=θ=02πr=15kr(z)z=02=r2-1drdθ

=0=02πr=15krr2-1drdθ

=θ=02πk2r=15(2r)r2-1drdθ

=θ=02πk2r2-13/23/25dθ

=8k3θ=02πdθ

Hence,Iz=16kπ3units

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