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Key equations

  • Mass of a lamina
    m = ρ a b f ( x ) d x
  • Moments of a lamina
    M x = ρ a b [ f ( x ) ] 2 2 d x and M y = ρ a b x f ( x ) d x
  • Center of mass of a lamina
    x = M y m and y = M x m

For the following exercises, calculate the center of mass for the collection of masses given.

m 1 = 2 at x 1 = 1 and m 2 = 4 at x 2 = 2

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m 1 = 1 at x 1 = −1 and m 2 = 3 at x 2 = 2

5 4

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m = 3 at x = 0 , 1 , 2 , 6

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Unit masses at ( x , y ) = ( 1 , 0 ) , ( 0 , 1 ) , ( 1 , 1 )

( 2 3 , 2 3 )

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m 1 = 1 at ( 1 , 0 ) and m 2 = 4 at ( 0 , 1 )

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m 1 = 1 at ( 1 , 0 ) and m 2 = 3 at ( 2 , 2 )

( 7 4 , 3 2 )

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For the following exercises, compute the center of mass x .

ρ = 1 for x ( −1 , 3 )

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ρ = x 2 for x ( 0 , L )

3 L 4

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ρ = 1 for x ( 0 , 1 ) and ρ = 2 for x ( 1 , 2 )

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ρ = sin x for x ( 0 , π )

π 2

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ρ = cos x for x ( 0 , π 2 )

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ρ = e x for x ( 0 , 2 )

e 2 + 1 e 2 1

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ρ = x 3 + x e x for x ( 0 , 1 )

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ρ = x sin x for x ( 0 , π )

π 2 4 π

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ρ = x for x ( 1 , 4 )

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ρ = ln x for x ( 1 , e )

1 4 ( 1 + e 2 )

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For the following exercises, compute the center of mass ( x , y ) . Use symmetry to help locate the center of mass whenever possible.

ρ = 7 in the square 0 x 1 , 0 y 1

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ρ = 3 in the triangle with vertices ( 0 , 0 ) , ( a , 0 ) , and ( 0 , b )

( a 3 , b 3 )

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ρ = 2 for the region bounded by y = cos ( x ) , y = cos ( x ) , x = π 2 , and x = π 2

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For the following exercises, use a calculator to draw the region, then compute the center of mass ( x , y ) . Use symmetry to help locate the center of mass whenever possible.

[T] The region bounded by y = cos ( 2 x ) , x = π 4 , and x = π 4

( 0 , π 8 )

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[T] The region between y = 2 x 2 , y = 0 , x = 0 , and x = 1

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[T] The region between y = 5 4 x 2 and y = 5

( 0 , 3 )

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[T] Region between y = x , y = ln ( x ) , x = 1 , and x = 4

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[T] The region bounded by y = 0 , x 2 4 + y 2 9 = 1

( 0 , 4 π )

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[T] The region bounded by y = 0 , x = 0 , and x 2 4 + y 2 9 = 1

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[T] The region bounded by y = x 2 and y = x 4 in the first quadrant

( 5 8 , 1 3 )

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For the following exercises, use the theorem of Pappus to determine the volume of the shape.

Rotating y = m x around the x -axis between x = 0 and x = 1

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Rotating y = m x around the y -axis between x = 0 and x = 1

m π 3

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A general cone created by rotating a triangle with vertices ( 0 , 0 ) , ( a , 0 ) , and ( 0 , b ) around the y -axis. Does your answer agree with the volume of a cone?

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A general cylinder created by rotating a rectangle with vertices ( 0 , 0 ) , ( a , 0 ) , ( 0 , b ) , and ( a , b ) around the y -axis. Does your answer agree with the volume of a cylinder?

π a 2 b

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A sphere created by rotating a semicircle with radius a around the y -axis. Does your answer agree with the volume of a sphere?

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For the following exercises, use a calculator to draw the region enclosed by the curve. Find the area M and the centroid ( x , y ) for the given shapes. Use symmetry to help locate the center of mass whenever possible.

[T] Quarter-circle: y = 1 x 2 , y = 0 , and x = 0

( 4 3 π , 4 3 π )

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[T] Triangle: y = x , y = 2 x , and y = 0

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[T] Lens: y = x 2 and y = x

( 1 2 , 2 5 )

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[T] Ring: y 2 + x 2 = 1 and y 2 + x 2 = 4

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[T] Half-ring: y 2 + x 2 = 1 , y 2 + x 2 = 4 , and y = 0

( 0 , 28 9 π )

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Find the generalized center of mass in the sliver between y = x a and y = x b with a > b . Then, use the Pappus theorem to find the volume of the solid generated when revolving around the y -axis.

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Find the generalized center of mass between y = a 2 x 2 , x = 0 , and y = 0 . Then, use the Pappus theorem to find the volume of the solid generated when revolving around the y -axis.

Center of mass: ( a 6 , 4 a 2 5 ) , volume: 2 π a 4 9

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Find the generalized center of mass between y = b sin ( a x ) , x = 0 , and x = π a . Then, use the Pappus theorem to find the volume of the solid generated when revolving around the y -axis.

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Use the theorem of Pappus to find the volume of a torus (pictured here). Assume that a disk of radius a is positioned with the left end of the circle at x = b , b > 0 , and is rotated around the y -axis.

This figure is a torus. It has inner radius of b. Inside of the torus is a cross section that is a circle. The circle has radius a.

Volume: 2 π 2 a 2 ( b + a )

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Find the center of mass ( x , y ) for a thin wire along the semicircle y = 1 x 2 with unit mass. ( Hint: Use the theorem of Pappus.)

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Practice Key Terms 6

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Source:  OpenStax, Calculus volume 1. OpenStax CNX. Feb 05, 2016 Download for free at http://cnx.org/content/col11964/1.2
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