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Consider the same region Q ( [link] ) and use the density function ρ ( x , y , z ) = x y 2 z . Find the center of mass.

( 3 2 , 9 8 , 1 2 )

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We conclude this section with an example of finding moments of inertia I x , I y , and I z .

Finding the moments of inertia of a solid

Suppose that Q is a solid region and is bounded by x + 2 y + 3 z = 6 and the coordinate planes with density ρ ( x , y , z ) = x 2 y z (see [link] ). Find the moments of inertia of the tetrahedron Q about the y z -plane, the x z -plane, and the x y -plane .

Once again, we can almost immediately write the limits of integration and hence we can quickly proceed to evaluating the moments of inertia. Using the formula stated before, the moments of inertia of the tetrahedron Q about the x y -plane, the x z -plane, and the y z -plane are

I x = Q ( y 2 + z 2 ) ρ ( x , y , z ) d V , I y = Q ( x 2 + z 2 ) ρ ( x , y , z ) d V ,

and

I z = Q ( x 2 + y 2 ) ρ ( x , y , z ) d V with ρ ( x , y , z ) = x 2 y z .

Proceeding with the computations, we have

I x = Q ( y 2 + z 2 ) x 2 y z d V = x = 0 x = 6 y = 0 y = 1 2 ( 6 x ) z = 0 z = 1 3 ( 6 x 2 y ) ( y 2 + z 2 ) x 2 y z d z d y d x = 117 35 3.343 , I y = Q ( x 2 + z 2 ) x 2 y z d V = x = 0 x = 6 y = 0 y = 1 2 ( 6 x ) z = 0 z = 1 3 ( 6 x 2 y ) ( x 2 + z 2 ) x 2 y z d z d y d x = 684 35 19.543 , I z = Q ( x 2 + y 2 ) x 2 y z d V = x = 0 x = 6 y = 0 y = 1 2 ( 6 x ) z = 0 z = 1 3 ( 6 x 2 y ) ( x 2 + y 2 ) x 2 y z d z d y d x = 729 35 20.829.

Thus, the moments of inertia of the tetrahedron Q about the y z -plane, the x z -plane, and the x y -plane are 117 / 35 , 684 / 35 , and 729 / 35 , respectively.

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Consider the same region Q ( [link] ), and use the density function ρ ( x , y , z ) = x y 2 z . Find the moments of inertia about the three coordinate planes.

The moments of inertia of the tetrahedron Q about the y z -plane, the x z -plane, and the x y -plane are 99 / 35 , 36 / 7 , and 243 / 35 , respectively.

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Key concepts

Finding the mass, center of mass, moments, and moments of inertia in double integrals:

  • For a lamina R with a density function ρ ( x , y ) at any point ( x , y ) in the plane, the mass is m = R ρ ( x , y ) d A .
  • The moments about the x -axis and y -axis are
    M x = R y ρ ( x , y ) d A and M y = R x ρ ( x , y ) d A .
  • The center of mass is given by x = M y m , y = M x m .
  • The center of mass becomes the centroid of the plane when the density is constant.
  • The moments of inertia about the x axis, y axis, and the origin are
    I x = R y 2 ρ ( x , y ) d A , I y = R x 2 ρ ( x , y ) d A , and I 0 = I x + I y = R ( x 2 + y 2 ) ρ ( x , y ) d A .

Finding the mass, center of mass, moments, and moments of inertia in triple integrals:

  • For a solid object Q with a density function ρ ( x , y , z ) at any point ( x , y , z ) in space, the mass is m = Q ρ ( x , y , z ) d V .
  • The moments about the x y -plane, the x z -plane, and the y z -plane are
    M x y = Q z ρ ( x , y , z ) d V , M x z = Q y ρ ( x , y , z ) d V , M y z = Q x ρ ( x , y , z ) d V .
  • The center of mass is given by x = M y z m , y = M x z m , z = M x y m .
  • The center of mass becomes the centroid of the solid when the density is constant.
  • The moments of inertia about the y z -plane, the x z -plane, and the x y -plane are
    I x = Q ( y 2 + z 2 ) ρ ( x , y , z ) d V , I y = Q ( x 2 + z 2 ) ρ ( x , y , z ) d V , I z = Q ( x 2 + y 2 ) ρ ( x , y , z ) d V .

Key equations

  • Mass of a lamina
    m = lim k , l i = 1 k j = 1 l m i j = lim k , l i = 1 k j = 1 l ρ ( x i j * , y i j * ) Δ A = R ρ ( x , y ) d A
  • Moment about the x -axis
    M x = lim k , l i = 1 k j = 1 l ( y i j * ) m i j = lim k , l i = 1 k j = 1 l ( y i j * ) ρ ( x i j * , y i j * ) Δ A = R y ρ ( x , y ) d A
  • Moment about the y -axis
    M y = lim k , l i = 1 k j = 1 l ( x i j * ) m i j = lim k , l i = 1 k j = 1 l ( x i j * ) ρ ( x i j * , y i j * ) Δ A = R x ρ ( x , y ) d A
  • Center of mass of a lamina
    x = M y m = R x ρ ( x , y ) d A R ρ ( x , y ) d A and y = M x m = R y ρ ( x , y ) d A R ρ ( x , y ) d A

In the following exercises, the region R occupied by a lamina is shown in a graph. Find the mass of R with the density function ρ .

Practice Key Terms 1

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Source:  OpenStax, Calculus volume 3. OpenStax CNX. Feb 05, 2016 Download for free at http://legacy.cnx.org/content/col11966/1.2
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