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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 cylindrical coordinates.

27. Find the mass and center of mass of the solid S hounded by the paraboloid z = 4x2 + 4y2 and the plane z = a (a >0) if S has constant density K.

To determine

To find: The mass and center of the given solid.

Explanation

Given:

The solid S is bounded by the paraboloid z=4x2+4y2 and the plane z=a(a>0) .

The density function is constant say K.

Calculation:

The mass of the given solid S is, m=EKf(x,y,z)dV . From the given conditions it is observed that, θ varies from 0 to 2π and z varies from 4r2 to a. To find the limits of r solve the given equations.

z=4x2+4y2a=4(r2)r2=a4r=a2

Therefore, the mass of the given solid is given by,

EKf(x,y,z)dV=02π0a24r2aK(r)dzdrdθ=K02π0a24r2ardzdrdθ

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

K02π0a24r2ardzdrdθ=K02π0a2r[z]4r2adrdθ=K02π0a2r(a4r2)drdθ=K02π0a2(ar4r3)drdθ

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

K02π0a24r2ardzdrdθ=K02π[ar224r44]0a2dt=K02π[(a(a2)22(a2)4)(a(0)22(0)4)]dθ=K02π[(a22(4)a216)(00)]dθ=K02π(a216)dθ

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

K02π0a24r2ardzdrdθ=Ka216[θ]02π=Ka216[2π0]=Ka216[2π]=Ka2π8

It is given that the density is constant and m=a2π8K , where K is the constant.

By the symmetric of the given function, it is observed that x and y coordinates of the center is 0. Thus, the third coordinate is given by,

z¯=02π0a24r2aKz(r)dzdrdθm=K02π0a24r2arzdzdrdθm

Integrate with respect to z and apply the limit of it

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