   Chapter 15.8, Problem 27E

Chapter
Section
Textbook Problem

Use spherical coordinates.27. Find the volume of the part of the ball ρ ≤ a that lies between the cones ϕ = π/6 and ϕ = π/3.

To determine

To find: The volume of the given region by using spherical coordinates.

Explanation

Given:

The region B part of the ball ρa that lies between the cones ϕ=π6 and ϕ=π3 .

Formula used:

The volume of the given region B is, BdV .

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 π6 to π3 . Use the formula mentioned above to find the volume of the given region

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