   Chapter 15.8, Problem 26E

Chapter
Section
Textbook Problem

Use spherical coordinates.26. Evaluate ∫∫∫E x 2 + y 2 + z 2 dV, where E lies above the cone z = x 2 + y 2 and between the spheres x2 + y2 + z2 = 1 and x2 + y2 + z2 = 4.

To determine

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

Explanation

Given:

The function is f(x,y,z)=x2+y2+z2 .

The region B lies above the cone z=x2+y2 and between the spheres x2+y2+z2=1 and x2+y2+z2=4 .

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 1 to 2, θ varies from 0 to 2π and ϕ varies from 0 to π4 . Use the formula mentioned above to change the given problem into spherical coordinates

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