   Chapter 14.6, Problem 48E

Chapter
Section
Textbook Problem

Centroid In Exercises 47-52, find the centroid of the solid region hounded by the graphs of the equations or described by the figure. Use a computer algebra system to evaluate the triple integrals. (Assume uniform density and find the center of mass.) y = 9 − x 2 ,       z = y ,       z = 0

To determine

To calculate: The centroid of the solid region bounded by graphs of the equations.

Explanation

Given:

The equations ρ(x,y,z)=k,Q:y=9x2,z=y,z=0.

Formula used:

Q=Qρ(x,y,z)dv

Calculation:

Total mass of the solid region

Q=Qρ(x,y,z)dvm=Qkdv=2k0309x20ydzdydx(1)

By using computer algebra system, we get the total mass of the solid

m=18k

The first moment in xz plane is

Myz=Qxρ(x,y,z)dvMyz=Qkxdv=k3309x20yxdzdydx(1)

By using computer algebra system, we get total mass of the solid

Myz=0

The first moment in xz plane is

Mxz=Qyρ(x,y,z)dvM

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