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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 x ¯ using ρ ( x , y , z ) = k .

Q : x + y + z = 10 , x = 0 , y = 0 , z = 0

To determine

To calculate: The mass and the indicated coordinate x¯ of center of mass of solid region Q with density given as (ρ=k). The solid region Q is bounded by the graph of the equation Q:x+y+z=10,x=0,y=0,z=0.

Explanation

Given:

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

Q:x+y+z=10,x=0,y=0,z=0.

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,

x¯=Myzm

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:x+y+z=10,x=0,y=0,z=0 into consideration.

The upper limit of x can be calculated from the above equation as follows:

x+y+z=10

If value of x and y are zero, then,

x+0+0=10x=10

Thus, the upper limit of x is 10 and bounds of x is 0x10.

Take the xy plane into consideration.

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

y=10x

Also, as z depends on both x and y, the equation of z contain both x and y

x+y+z=10

On subtracting x+y from both sides, we get,

z=10xy

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=010010x010xykdzdydx=k010010x[z]010xydydx=k010010x[(10xy)0]dydx

Next, on integrating with respect to y, we get,

k010010x[(10xy)0]dydx=k010[10yxyy22]010xdx=ky010[10xy2]010xdx=k(10x)010[10x(10x)20]dx=k(10x)010[(10x)(10x)20]dx

Simplifying further,

k010010x[(10xy)0]dydx=k(10x)010[(10x)2]dx=k2010(10x)2dx=k2010(100+x220x)dx

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

k2010(100+x220x)dx=k2[100x+x3320x22]010=k2[100(10)+103310(10)20]=k2[1000+100031000]=5003k

Therefore, mass (m) of the solid body Q is calculated to be 5003k

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