   Chapter 15.8, Problem 21E

Chapter
Section
Textbook Problem

Use spherical coordinates.21. Evaluate ∫∫∫B (x2+y2 + z2)2 dV, where B is the ball with center the origin and radius 5.

To determine

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

Explanation

Given:

The function is f(x,y,z)=(x2+y2+z2)2 .

The region B is the ball with center at origin and radius 5.

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 5, θ varies from 0 to 2π and ϕ varies from 0 to π .

Use the formula mentioned above to change the given problem into spherical coordinates

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