Chapter 6: Applications of Double Integration
Section 6.5: First Moments
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Example 6.5.7
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Calculate the coordinates of the center of mass of the lamina whose density is , and whose shape is that of , the region bounded by and .
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Solution
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Mathematical Solution
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Figure 6.5.7(a) shows the region in red, the center of mass (green dot), and the density as a surface in blue. The relevant calculations are
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Figure 6.5.7(a) CM, , and
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Maple Solution - Interactive
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Obtain the intersections of the bounding curves
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Write a sequence of two equations and press the Enter key.
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Context Panel: Solve≻Solve
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A solution from first principles is detailed in Table 6.5.7(a).
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Define the density function
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Context Panel: Assign Name
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Obtain , the total mass of the lamina
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Calculus palette:
Iterated double-integral template
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Context Panel: Evaluate and Display Inline
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Context Panel: Assign to a Name≻
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Obtain , the total moments about the -axis
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Calculus palette:
Iterated double-integral template
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Context Panel: Evaluate and Display Inline
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Context Panel: Assign to a Name≻
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Obtain , the total moments about the -axis
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Calculus palette:
Iterated double-integral template
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Context Panel: Evaluate and Display Inline
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Context Panel: Assign to a Name≻
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Obtain
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Context Panel: Evaluate and Display Inline
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Obtain
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Context Panel: Evaluate and Display Inline
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Table 6.5.7(a) Calculation of the center of mass from first principles
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Maple Solution - Coded
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A solution from first principles is provided in Table 6.5.7(b).
Initialize
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Define the density function .
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Obtain the total mass of the lamina
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Display the unevaluated integral with the Int command, and evaluate the integral with the value command.
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Obtain the first moments and
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Display the unevaluated integral with the Int command, and evaluate the integral with the value command.
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Obtain the coordinates of the center of mass
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Implement the relevant arithmetic.
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Table 6.5.7(b) Coordinates of the center of mass calculated from first principles
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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
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Obtain the coordinates of the center of mass
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Obtain a graph of the region , the density , and the center of mass
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Table 6.5.7(c) Coordinates of the center of mass calculated with the CenterOfMass command
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