Chapter 6: Applications of Double Integration
Section 6.5: First Moments
Calculate the coordinates of the center of mass of the lamina whose density is ρ=1+2 x2+3 y2/10, and whose shape is that of R, the region bounded by x=y2 and y=x−2.
Figure 6.5.7(a) shows the region R in red, the center of mass (green dot), and the density ρ as a surface in blue. The relevant calculations are
m=∫−12∫y2y+2ρ ⅆx ⅆy = 61831400
Mx=∫−12∫y2y+2ρ⋅y ⅆx ⅆy = 10825
My=∫−12∫y2y+2ρ⋅x ⅆx ⅆy = 7263700
Figure 6.5.7(a) CM, R, and ρ
Maple Solution - Interactive
Obtain the intersections of the bounding curves
Write a sequence of two equations and press the Enter key.
Context Panel: Solve≻Solve
A solution from first principles is detailed in Table 6.5.7(a).
Define the density function ρx,y
Context Panel: Assign Name
ρ=1+2 x2+3 y2/10→assign
Obtain m, the total mass of the lamina
Iterated double-integral template
Context Panel: Evaluate and Display Inline
Context Panel: Assign to a Name≻m
∫−12∫y2y+2ρ ⅆx ⅆy = 61831400→assign to a namem
Obtain Mx, the total moments about the x-axis
Context Panel: Assign to a Name≻Mx
∫−12∫y2y+2ρ⋅y ⅆx ⅆy = 10825→assign to a nameMx
Obtain My, the total moments about the y-axis
Context Panel: Assign to a Name≻My
∫−12∫y2y+2ρ⋅x ⅆx ⅆy = 7263700→assign to a nameMy
My/m = 538229
Mx/m = 224229
Table 6.5.7(a) Calculation of the center of mass from first principles
Maple Solution - Coded
A solution from first principles is provided in Table 6.5.7(b).
Define the density function ρx,y.
ρ≔1+2 x2+ 3 y2/10:
Obtain the total mass of the lamina
Display the unevaluated integral with the Int command, and evaluate the integral with the value command.
Obtain the first moments Mx and My
Obtain the coordinates of the center of mass x&conjugate0;,y&conjugate0;
Implement the relevant arithmetic.
Mym,Mxm = 538229,224229
Table 6.5.7(b) Coordinates of the center of mass calculated from first principles
In Table 6.5.7(c), the coordinates of the center of mass are calculated using the CenterOfMass command from the Student MultivariateCalculus package.
Obtain the inert integrals defining the center of mass
Obtain the coordinates of the center of mass
Obtain a graph of the region R, the density ρ, and the center of mass
Table 6.5.7(c) Coordinates of the center of mass calculated with the CenterOfMass command
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