   Chapter 15.8, Problem 42E

Chapter
Section
Textbook Problem

Evaluate the integral by changing to spherical coordinates.42.   ∫ − a a ∫ − a 2 − y 2 a 2 − y 2 ∫ − a 2 − x 2 − y 2 a 2 − x 2 − y 2 (x2z + y2z + z3) dz dx dy

To determine

To evaluate: The integral by changing to spherical coordinates.

Explanation

Given:

The function is f(x,y,z)=x2z+y2z+z3 .

The rectangular coordinates of the given triple integral are {(x,y,z)|aya,a2y2xa2y2,a2x2y2za2x2y2} .

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:

Substitute x=ρsinϕcosθ,y=ρsinϕsinθ,z=ρcosϕ in the given function f(x,y,z) .

f(x,y,z)=x2z+y2z+z3f(ρ,θ,ϕ)=ρ2sin2ϕcos2θρcosϕ+ρ2sin2ϕsin2θρcosϕ+ρ3cos3ϕf(ρ,θ,ϕ)=ρ3sin2ϕcosϕ+ρ3cos3ϕf(ρ,θ,ϕ)=ρ3cosϕ(sin2ϕ+cos2ϕ)

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