   Chapter 15.8, Problem 22E

Chapter
Section
Textbook Problem

Use spherical coordinates.22. Evaluate ∫∫∫E y2z2 dV, where E lies above the cone ϕ = π/3 and below the sphere ρ = 1.

To determine

To evaluate: The given triple integral by using spherical coordinates.

Explanation

Given:

The function is f(x,y,z)=y2z2 .

The region E lies above the cone ϕ=π3 and below the sphere ρ=1 .

Formula used:

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 1, θ varies from 0 to 2π and ϕ varies from 0 to π3 .

Use the formula mentioned above to change the given problem into spherical coordinates. Then, by the equation (1), the value of the given triple integral is,

Bf(x,y,z)dV=0102π0π3y2z2dzdydx=0502π0π3(ρsinϕsinθ)2(ρcosϕ)2(ρ2sinϕ)dϕdθdρ=0502π0π3(ρ2sin2ϕsin2θ)(ρ2cos2ϕ)(ρ2sinϕ)dϕdθdρ=0502π0π3ρ6sin3ϕcos2ϕsin2θdϕdθdρ

Use the equation (2) to integrate and apply the limit values

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