   Chapter 15.8, Problem 24E

Chapter
Section
Textbook Problem

Use spherical coordinates.24. Evaluate ∫∫∫E y2 dV, where E is the solid hemisphere x2 + y2 + z2 ≤ 9, y ≥ 0.

To determine

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

Explanation

Given:

The function is f(x,y,z)=y2 .

The region B is hemisphere x2+y2+z29 and y0 .

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 3, θ varies from 0 to π and ϕ varies from 0 to π . Use the formula mentioned above to change the given problem into spherical coordinates. Then, by equation (1), the value of the given triple integral is,

Bf(x,y,z)dV=030π0πy2dzdydx=030π0π(ρsinϕsinθ)2(ρ2sinϕ)dϕdθdρ=030π0π[ρ2sin2ϕsin2θ](ρ2sinϕ)dϕdθdρ=030π0πρ4sin3ϕsin2θdϕdθdρ

Use (2) to integrate and apply the limit values

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