Chapter 8: Applications of Triple Integration
Section 8.3: First Moments
Example 8.3.3
Obtain the centroid of R, the region common to the two cylinders x2+y2=1 and x2+z2=1.
Impose the density δx,y,z=3+x+y+z on R and find the resulting center of mass.
(See Example 8.1.8.)
Solution
Mathematical Solution
The volume of R, found in Example 8.1.8, is given by
V=∫−11∫−1−x21−x2∫−1−x21−x21 dz dy dx = 163
Table 8.3.3(a) lists the first moments and the coordinates of the centroid.
First Moments
Centroid
Myz=∫−11∫−1−x21−x2∫−1−x21−x2x dz dy dx=0
x&conjugate0;=MyzV=0V=0
Mxz=∫−11∫−1−x21−x2∫−1−x21−x2y dz dy dx=0
y&conjugate0;=MxzV=0V=0
Mxy=∫−11∫−1−x21−x2∫−1−x21−x2z dz dy dx=0
z&conjugate0;=MxyV=0V=0
Table 8.3.3(a) First moments and the coordinates of the centroid
When the region R supports the density δx,y,z=3+x+y+z, the total mass in R is
m=∫−11∫−1−x21−x2∫−1−x21−x23+x+y+z dz dy dx = 16
Table 8.3.3(b) lists the first moments and the coordinates of the center of mass under this condition.
Center of Mass
Myz=∫−11∫−1−x21−x2∫−1−x21−x2x3+x+y+z dz dy dx = 1615
x&conjugate0;=Myzm=161516 = 115
Mxz=∫−11∫−1−x21−x2∫−1−x21−x2y3+x+y+z dz dy dx = 6445
y&conjugate0;=Mxzm=644516 = 445
Mxy=∫−11∫−1−x21−x2∫−1−x21−x2z3+x+y+z dz dy dx = 6445
z&conjugate0;=Mxym=644516 = 445
Table 8.3.3(b) First moments and the coordinates of the center of mass
Maple Solution - Interactive
Based on the CenterOfMass command in the Student MultivariateCalculus package, the task template in Table 8.3.3(c) will find the centroid of R when the density is set to 1.
Tools≻Tasks≻Browse: Calculus - Multivariate≻Integration≻Center of Mass≻Cartesian 3-D
Center of Mass for 3D Region in Cartesian Coordinates
Density:
1
Region: z1x,y≤z≤z2x,y,y1x≤y≤y2x,a≤x≤b
z1x,y
−1−x2
−−x2+1
z2x,y
1−x2
−x2+1
y1x
y2x
a
−1
b
Moments÷Mass:
Inert Integral - dz dy dx
StudentMultivariateCalculusCenterOfMass,z=.., y=..,x=..,output=integral
∫−11∫−−x2+1−x2+1∫−−x2+1−x2+1xⅆzⅆyⅆx∫−11∫−−x2+1−x2+1∫−−x2+1−x2+11ⅆzⅆyⅆx,∫−11∫−−x2+1−x2+1∫−−x2+1−x2+1yⅆzⅆyⅆx∫−11∫−−x2+1−x2+1∫−−x2+1−x2+11ⅆzⅆyⅆx,∫−11∫−−x2+1−x2+1∫−−x2+1−x2+1zⅆzⅆyⅆx∫−11∫−−x2+1−x2+1∫−−x2+1−x2+11ⅆzⅆyⅆx
Explicit values for x&conjugate0;, y&conjugate0;, and z&conjugate0;
StudentMultivariateCalculusCenterOfMass,z=.., y=..,x=..
0,0,0
Table 8.3.3(c) Centroid computed by task template that implements the CenterOfMass command
Based on the CenterOfMass command in the Student MultivariateCalculus package, the task template in Table 8.3.3(d) will find the center of mass of R for a given density.
3+x+y+z
∫−11∫−−x2+1−x2+1∫−−x2+1−x2+1x3+x+y+zⅆzⅆyⅆx∫−11∫−−x2+1−x2+1∫−−x2+1−x2+13+x+y+zⅆzⅆyⅆx,∫−11∫−−x2+1−x2+1∫−−x2+1−x2+1y3+x+y+zⅆzⅆyⅆx∫−11∫−−x2+1−x2+1∫−−x2+1−x2+13+x+y+zⅆzⅆyⅆx,∫−11∫−−x2+1−x2+1∫−−x2+1−x2+1z3+x+y+zⅆzⅆyⅆx∫−11∫−−x2+1−x2+1∫−−x2+1−x2+13+x+y+zⅆzⅆyⅆx
115,445,445
Table 8.3.3(d) Centroid computed by task template that implements the CenterOfMass command
Maple Solution - Coded
In Table 8.3.3(e), the centroid of R is obtained via the CenterOfMass command from the Student MultivariateCalculus package, provided the density is set equal to 1.
Initialize
Install the Student MultivariateCalculus package.
withStudent:-MultivariateCalculus:
Context Panel: Assign Name
g=1−x2→assign
Apply the CenterOfMass command from the Student MultivariateCalculus package
CenterOfMass1,z=−g..g,y=−g..g,x=−1..1,output=integral
C≔CenterOfMass1,z=−g..g,y=−g..g,x=−1..1
Table 8.3.3(e) Centroid in Cartesian coordinates
In Table 8.3.3(f), the center of mass is obtained via the CenterOfMass command from the Student MultivariateCalculus package.
Define the density δ.
δ≔3+x+y+z:
CenterOfMassδ,z=−g..g,y=−g..g,x=−1..1,output=integral
CM≔CenterOfMassδ,z=−g..g,y=−g..g,x=−1..1
Table 8.3.3(f) Center of mass in Cartesian coordinates
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