Chapter 6: Applications of Double Integration
Section 6.5: First Moments
Essentials
The coordinates x&conjugate0;,y&conjugate0; of the center of mass of a plane lamina occupying a region R and having density ρx,y are given by
x&conjugate0;=Mym and y&conjugate0;=Mxm
where
m= ∫∫Rρ dA is the total mass
My= ∫∫Rρ x dA is the total first moment about the y-axis
Mx= ∫∫Rρ y dA is the total first moment about the x-axis
The center of mass is the point where, should all the mass be concentrated, the moments about the axes would be the same as for the lamina.
If ρ=1, then x&conjugate0;,y&conjugate0;is the centroid, or geometric center of R.
Maple Tools for Calculating a Center of Mass
The CenterOfMass command in the Student MultivariateCalculus package is the primary tool for computing a centroid or a center of mass. This command will return the inert integrals defining x&conjugate0; and y&conjugate0;, or the value of these integrals, or, for a lamina, a graph displaying the region R, the surface ρ, and the point x&conjugate0;,y&conjugate0;.
There are also two relevant task templates that implement the CenterOfMass command, one each for a double integral in Cartesian or polar coordinates. However, these task templates iterate in a fixed order: dy dx for the Cartesian case; and dr dθ for the polar.
Of course, the MultiInt command in the Student MultivariateCalculus package (or the modified int command in the Student VectorCalculus package) can be used to integrate over triangles, or over all or part of disks and ellipses. For each such region, a robust task template implements the underlying command, but the MultiInt command can be more easily implemented from the Context Panel when the Student MultivariateCalculus package has been installed.
Examples
Example 6.5.1
Determine the coordinates of the centroid of R, the region bounded by y=1−x2 and the x-axis.
Example 6.5.2
Determine the coordinates of the center of mass of a lamina in the shape of R in Example 6.5.1 if its density is equal to the square of the distance from the point 0,1/2.
Example 6.5.3
If a,b,c are positive, and a>b, determine the coordinates of the centroid of the triangle whose vertices are 0,0,a,0,b,c.
Example 6.5.4
Calculate the coordinates of the center of mass for the triangular lamina whose vertices are 0 ,0,0,3,4,0, and whose density is ρ=1+2 x+ y/2.
Example 6.5.5
Determine the coordinates of the center of mass for a semicircular lamina whose density is equal to the distance from the center of the circle, and whose radius is 1.
Example 6.5.6
Calculate the coordinates of the center of mass of the lamina whose shape is that of the cardioid r=1+cosθ and whose density is ρ=1+2 x2+3 y2/10.
Example 6.5.7
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.
Example 6.5.8
Calculate the coordinates of the center of mass of the lamina whose shape is that of an isosceles right triangle with legs of length 1, and whose density is twice the square of the distance from the vertex of the right angle. Hint: Put the right angle at the origin.
Example 6.5.9
Calculate the coordinates of the centroid of R, the region bounded by the graphs of x=0, and x=y−y3, if y∈0,1.
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.
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