   Chapter 15.8, Problem 25E

Chapter
Section
Textbook Problem

Use spherical coordinates.25. Evaluate ∫∫∫E xe x2 + y2 + z2 dV, where E is the position of the unit ball x2 + y2 + z2 ≤ 1 that lies in the first octant.

To determine

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

Explanation

Given:

The function is f(x,y,z)=xex2+y2+z2 .

The region B is part of the sphere x2+y2+z21 lies in the first quadrant.

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 π2 and ϕ varies from 0 to π2 . 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=010π20π2xex2+y2+z2dzdydx=010π20π2(ρsinϕcosθ)eρ2(ρ2sinϕ)dϕdθdρ=010π20π2ρ3eρ2sin2ϕcosθdϕdθdρ

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

010π20π2ρ3eρ2sin2ϕcosθdϕdθdρ=01ρ3eρ2dρ0π2cosθdθ0π2sin2ϕdϕ=01ρ3eρ2dρ0π2cosθdθ0π2(1cos2ϕ2)sinϕdϕ=1201ρ3eρ2dρ0π2cosθdθ0π2(1cos2ϕ)sinϕdϕ

Substitute t=ρ2,dt=2ρdρ

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