Chapter 6: Applications of Double Integration
Section 6.5: First Moments
Example 6.5.10
Determine the coordinates of the center of mass of the lamina whose density is ρ=3+2 x+3 y, and whose shape is that of R, the region bounded by the ellipse x2+4 y2=1.
Solution
Mathematical Solution
Figure 6.5.10(a) shows the region R in red, and the surface ρ, in blue. The green dot represents the center of mass x&conjugate0;,y&conjugate0;=1/6,1/16. The relevant calculations are tabulated to the left of the figure, where ρ=3+r2 cosθ+3 sinθ and φθ=1/4−3 cos2θ.
m=∫02 π∫0φθr⋅ρ ⅆr ⅆθ = 32 π
Mx=∫02 π∫0φθr2⋅ρ sinθ ⅆr ⅆθ = 332 π
My=∫02 π∫0φθr2⋅ρ cosθ ⅆr ⅆθ = π4
x&conjugate0;=My/m=1/6
y&conjugate0;=Mx/m=1/16
Figure 6.5.10(a) Center of mass, R, and ρ
Maple Solution - Interactive
The total mass is calculated in Table 6.5.10(a) via a task template that integrates over an ellipse via the modified int command in the Student VectorCalculus package.
Tools≻Tasks≻Browse:
Calculus - Vector≻Integration≻Multiple Integration≻2-D≻Over an Ellipse
Integrate fx,y over an Ellipse
fx,y=
Equation of ellipse:
From θ= to θ=
Table 6.5.10(a) Total mass calculated with a task template that integrates over an ellipse
The ellipse is expressed in polar coordinates as
12tanθ2+114+tanθ2
=sec2θ1+4 tan2θ
=1/cos2θ+4 sinθ2
=1/cos2θ+41−cos2θ
=1/4−3 cos2θ
Similarly, the first moments Mx=3 π/32 and My=π/4 could be obtained with this same task template. Alternatively, as shown in Table 6.5.10(b), calculate the coordinates of the center of mass via the task template that implements (in polar coordinates) the CenterOfMass command from the Student MultivariateCalculus package. The density is then ρ=3+r 2 cosθ+3 sinθ.
Calculus - Multivariate≻Integration≻Center of Mass≻Polar
Center of Mass for Planar Region in Polar Coordinates
Density:
3+r 2 cosθ+3 sinθ
3+r2cosθ+3sinθ
Region: r1θ≤r ≤r2θ,a≤θ≤b
r1θ
0
r2θ
1/4−3 cos2θ
14−3cosθ2
a
b
2 π
2π
Moments÷Mass:
Inert Integral - dr dθ
StudentMultivariateCalculusCenterOfMass, r=..,θ=..,coordinates=polarr, θ,output=integral
∫02π∫014−3cosθ2cosθr23+r2cosθ+3sinθⅆrⅆθ∫02π∫014−3cosθ2r3+r2cosθ+3sinθⅆrⅆθ,∫02π∫014−3cosθ2sinθr23+r2cosθ+3sinθⅆrⅆθ∫02π∫014−3cosθ2r3+r2cosθ+3sinθⅆrⅆθ
Explicit values for r&conjugate0; and θ&conjugate0;
StudentMultivariateCalculusCenterOfMass,r=..,θ=..,coordinates=polarr, θ
14873,arctan38
Plot:
StudentMultivariateCalculusCenterOfMass,r=..,θ=.., coordinates=polarr, θ,output=plot,caption=
Table 6.5.10(b) Calculation of center of mass via task template
The figure produced by the option "output = plot" has had constrained scaling imposed via the Context Panel for the graph. The graph itself shows the region R in red, and the function ρ=r in blue. The centroid is represented by the green dot.
The polar coordinates of the centroid are therefore r&conjugate0;,θ&conjugate0;=, which, in Cartesian coordinates would be r&conjugate0; cosθ&conjugate0;,r&conjugate0; sinθ&conjugate0;=1/6,1/16. In other words, the task template calculates the mass and first moments in Cartesian coordinates, but returns the center of mass in polar coordinates.
Maple Solution - Coded
Initialize
Tools≻Load Package: Student Multivariate Calculus
Loading Student:-MultivariateCalculus
Define the density function ρ.
ρ≔3+2 x+3 y:
Obtain m, the total mass
Use the MultiInt command in the Student MultivariateCalculus package.
MultiIntρ,x,y=Ellipsex2+4 y2=1,output=integral;m≔MultiIntρ,x,y=Ellipsex2+4 y2=1
32π
Obtain Mx and My, the first moments
MultiInty⋅ρ,x,y=Ellipsex2+4 y2=1,output=integral;Mx≔MultiInty⋅ρ,x,y=Ellipsex2+4 y2=1
332π
MultiIntx⋅ρ,x,y=Ellipsex2+4 y2=1,output=integral;My≔MultiIntx⋅ρ,x,y=Ellipsex2+4 y2=1
14π
Obtain x&conjugate0;,y&conjugate0;, the coordinates of the center of mass
Mym,Mxm = 16,116
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