   Chapter 15.7, Problem 28E

Chapter
Section
Textbook Problem

Use cylindrical coordinates.28. Find the mass of a ball B given by x2 + y2 + z2 ≤ a2 if the density at any point is proportional to its distance from the z-axis.

To determine

To find: The mass of the given ball.

Explanation

Given:

The ball B is x2+y2+z2a2 .

The density function is proportional to the distance from the z-axis that is ρ(x,y,z)=Kx2+y2 .

Calculation:

The mass of the given solid S is, m=EKx2+y2f(x,y,z)dV . From the given conditions it is observed that, θ varies from 0 to 2π and r varies from 0 to a. To find the limits of z solve the given equations.

x2+y2+z2=a2r2+z2=a2z2=a2r2z=±a2r2

Therefore, the mass of the given solid is given by,

EKx2+y2f(x,y,z)dV=02π0aa2r2a2r2Kr2(r)dzdrdθ=2K02π0a0a2r2r(r)dzdrdθ=2K02π0a0a2r2r2dzdrdθ

Integrate with respect to z and apply the limit of it.

2K02π0a0a2r2r2dzdrdθ=2K02π0ar2[z]0a2r2drdθ=2K02π0ar2(a2r20)drdθ=2K02π0ar2a2r2drdθ

Separate the integrals, and integrate with respect to θ .

2K02π0a0a2r2r2dzdrdθ=2K0ar2a2r2dr02πdθ=2K0ar2a2r2dr[θ]02π=2K0ar2a2r2dr[2π0]=2(2π)K0ar2a2r2dr

Integrate with respect to r and apply the limit of it

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