   Chapter 15.8, Problem 28E

Chapter
Section
Textbook Problem

Use spherical coordinates.28. Find the average distance from a point in a ball of radius a to its center.

To determine

To find: The average distance of the given point from the center by using spherical coordinates.

Explanation

Given:

The region B is the ball of radius a.

Formula used:

The average distance of the given pointis, 1V(B)BρdV where V(B) is the volume of the given region B.

If f is a spherical region E given by aρb,αθβ,cϕd , then, Ef(x,y,z)dV=αβabcdf(ρsinϕcosθ,ρsinϕsinθ,ρcosϕ)ρ2sinϕdϕdρdθ (1)

If g(x) is the function of x and h(y) is the function of y and k(z) is the function of z  then, abcdefg(x)h(y)k(z)dzdydx=abg(x)dxcdh(y)dyefk(z)dz (2)

The spherical coordinates (ρ,θ,ϕ) corresponding to the rectangular coordinates (x,y,z) is,

ρ=x2+y2+z2ϕ=cos1(zρ)θ=cos1(xρsinϕ)

Calculation:

By the given conditions, it is observed that ρ varies from 0 to a, θ varies from 0 to 2π and ϕ varies from 0 to π . Use the formula mentioned above to find the average distance of the given point from the center. The given region is the ball of radius a. So, the volume of the region is given by 43πa3

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