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Calculus (MindTap Course List)

11th Edition
Ron Larson + 1 other
ISBN: 9781337275347

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BuyFindarrow_forward

Calculus (MindTap Course List)

11th Edition
Ron Larson + 1 other
ISBN: 9781337275347
Textbook Problem

Center of Mass In Exercises 59 and 60, find the mass and the indicated coordinate of the center of mass of the solid region Q of density ρ bounded by the graphs of the equations.

Find y ¯ using ρ ( x , y , z ) = k x .

Q : z = 5 y , z = 0 , y = 0 , x = 0 , x = 5

To determine

To calculate: The mass and the indicated coordinate y¯ of center of mass of solid region Q with density given as (ρ=kx). The solid region Q is bounded by the graph of equation Q:z=5y,x=0,y=0,z=0,x=5.

Explanation

Given:

The density of solid region Q is ρ(x,y,z)=kx and the solid is bounded by the graph of equation,

Q:z=5y,x=0,y=0,z=0,x=5.

Formula used:

Mass (m) of a solid region Q having density ρ is,

m=Qρ(x,y,z)dV

First moment about yz plane is,

Myz=Qxρ(x,y,z)dV

x Coordinate (x¯) of center of mass of a solid region Q is,

y¯=Mxzm

Calculation:

A 3-D plot is made with the help of maple software,

The limits of x,y and z can be calculated with the help of this 3-D plot.

Take the provided equation of the solid Q:z=5y,x=0,y=0,z=0,x=5 into consideration.

In the above equation, the bounds of x is 0x5 and bounds of z is, 0z(5y)

Take the xy plane into consideration.

In this case, since the value of z=0, the equation of the line will be given as,

0=5yy=5

The lower limit of x,y,z is provided in the question, i.e.,

x=0,y=0,z=0

On applying the formula for the mass (m) of solid body Q, we get,

m=Qρ(x,y,z)dV=050505ykxdzdydx=kx0505[z]05ydydx=kx0505[(5y)0]dydx

On integrating with respect to y, we get,

kx0505(5y)dydx=kx05[5yy22]05dx=kx05[5(5)5220]dx=252k05xdx

Further, on integrating with respect to x, we get,

252k05xdx=252k[x22]05=252k[5220]=6254k

Thus, mass (m) of the solid body Q is, 6254k

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Chapter 14 Solutions

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