   Chapter 15.8, Problem 35E

Chapter
Section
Textbook Problem

Use cylindrical or spherical coordinates, whichever seems more appropriate.35. Find the volume and centroid of the solid E that lies above the cone z = x 2 + y 2 and below the sphere x2 + y2 + z2 = 1.

To determine

To find: The volume and centroid of the given region by using spherical or cylindrical coordinates.

Explanation

Given:

The region B lies above the cone z=x2+y2 and below the sphere x2+y2+z2=1 .

Formula used:

The volume of the given region B is, BdV .

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:

Since the equation of the sphere and cone is given, it is more appropriate to use spherical coordinates. By the given conditions, it is observed that ρ varies from 0 to1, θ varies from 0 to 2π and to find the limits of ϕ , solve the given equation as below.

z=x2+y2ρcosϕ=ρ2sin2ϕcos2θ+ρ2sin2ϕsin2θρcosϕ=ρsinϕcos2θ+sin2θρcosϕ=ρsinϕ

So, ϕ varies from 0 to π4 . Use the formula mentioned above to find the volume of the given region. Then, by equation (1), the volume of the given region is,

BdV=02π0π401dzdydx=02π0π401(ρ2sinϕ)dρdϕdθ

Integrate with respect to ρ and apply the limit of it.

02π0π401(ρ2sinϕ)dρdϕdθ=02π0π4sinϕ[ρ33]01dϕdθ=02π0π4sinϕ[(1)33(0)33]dϕdθ=1302π0π4sinϕdϕdθ=1302π0π4sinϕdϕdθ

Use (2) to integrate and apply the limit values

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