   Chapter 15.8, Problem 29E

Chapter
Section
Textbook Problem

Use spherical coordinates.29. (a) Find the volume of the solid that lies above the cone ϕ = π/3 and below the sphere ρ = 4 cos ϕ.(b) Find the centroid of the solid in part (a).

(a)

To determine

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

Explanation

Given:

The region B lies above the cone ϕ=π3 and below the sphere ρ=4cosϕ .

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 4cosϕ , θ varies from 0 to 2π and ϕ varies from 0 to π3 . Use the formula mentioned above to find the volume of the given region. Then, by equation (1), the volume of the given region is,

BdV=02π0π304cosϕdzdydx=02π0π304cosϕ(ρ2sinϕ)dρdϕdθ

Integrate with respect to ρ and apply the limit of it

(b)

To determine

To find: The centroid of the solid in part (a).

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