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Calculus: Early Transcendentals

8th Edition
James Stewart
ISBN: 9781285741550

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BuyFindarrow_forward

Calculus: Early Transcendentals

8th Edition
James Stewart
ISBN: 9781285741550
Textbook Problem

Use spherical coordinates.

31. (a) Find the centroid of the solid in Example 4. (Assume constant density K.)

(b) Find the moment of inertia about the z-axis for this solid.

(a)

To determine

The centroid of the solid in example 4 by using spherical coordinates.

Explanation

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:

From example 4, it is known that m=π8K. From the symmetry of the given region, it is observed that the Myz and Mxz are equal to 0. The value of Mxy is given by,

Mxy=02π0π40cosϕKzdzdydx=K02π0π40cosϕρcosϕ(ρ2sinϕ)dρdϕdθ=K02π0π40cosϕρ3cosϕsinϕdρdϕdθ

Integrate with respect to ρ and apply the limit of it

(b)

To determine

The moment of inertia about the z axis.

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Chapter 15 Solutions

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