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R is the triangular region with vertices ( 0 , 0 ) , ( 0 , 3 ) , and ( 6 , 0 ) ; ρ ( x , y ) = x y .

A right triangle bounded by the x and y axes and the line y = negative x/2 + 3.

27 2

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R is the triangular region with vertices ( 0 , 0 ) , ( 1 , 1 ) , ( 0 , 5 ) ; ρ ( x , y ) = x + y .

A triangle bounded by the y axis, the line x = y, and the line y = negative 4x + 5.
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R is the rectangular region with vertices ( 0 , 0 ) , ( 0 , 3 ) , ( 6 , 3 ) , and ( 6 , 0 ) ; ρ ( x , y ) = x y .

A rectangle bounded by the x and y axes and the lines x = 6 and y = 3.

24 2

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R is the rectangular region with vertices ( 0 , 1 ) , ( 0 , 3 ) , ( 3 , 3 ) , and ( 3 , 1 ) ; ρ ( x , y ) = x 2 y .

A rectangle bounded by the y axis, the lines y = 1 and 3, and the line x = 3.
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R is the trapezoidal region determined by the lines y = 1 4 x + 5 2 , y = 0 , y = 2 , and x = 0 ; ρ ( x , y ) = 3 x y .

A trapezoid bounded by the x and y axes, the line y = 2, and the line y = negative x/4 + 2.5.

76

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R is the trapezoidal region determined by the lines y = 0 , y = 1 , y = x , and y = x + 3 ; ρ ( x , y ) = 2 x + y .

A trapezoid bounded by the x axis, the line y = 1, the line y = x, and the line y = negative x + 3.
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R is the disk of radius 2 centered at ( 1 , 2 ) ; ρ ( x , y ) = x 2 + y 2 2 x 4 y + 5 .

A circle with radius 2 centered at (1, 2), which is tangent to the x axis at (1, 0).

8 π

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R is the unit disk; ρ ( x , y ) = 3 x 4 + 6 x 2 y 2 + 3 y 4 .

A circle with radius 1 and center the origin.
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R is the region enclosed by the ellipse x 2 + 4 y 2 = 1 ; ρ ( x , y ) = 1 .

An ellipse with center the origin, major axis 2, and minor axis 0.5.

π 2

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R = { ( x , y ) | 9 x 2 + y 2 1 , x 0 , y 0 } ; ρ ( x , y ) = 9 x 2 + y 2 .

The quarter section of an ellipse in the first quadrant with center the origin, major axis 2, and minor axis roughly 0.64.
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R is the region bounded by y = x , y = x , y = x + 2 , y = x + 2 ; ρ ( x , y ) = 1 .

A square with side length square root of 2 rotated 45 degrees, with corners at the origin, (2, 0), (1, 1), and (negative 1, 1).

2

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R is the region bounded by y = 1 x , y = 2 x , y = 1 , and y = 2 ; ρ ( x , y ) = 4 ( x + y ) .

A complex region between 2 and 1 that sweeps down and to the right with boundaries y = 1/x and y = 2/x.
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In the following exercises, consider a lamina occupying the region R and having the density function ρ given in the preceding group of exercises. Use a computer algebra system (CAS) to answer the following questions.

  1. Find the moments M x and M y about the x -axis and y -axis, respectively.
  2. Calculate and plot the center of mass of the lamina.
  3. [T] Use a CAS to locate the center of mass on the graph of R .

[T] R is the triangular region with vertices ( 0 , 0 ) , ( 0 , 3 ) , and ( 6 , 0 ) ; ρ ( x , y ) = x y .

a. M x = 81 5 , M y = 162 5 ; b. x = 12 5 , y = 6 5 ;
c.
A triangular region R bounded by the x and y axes and the line y = negative x/2 + 3, with a point marked at (12/5, 6/5).

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[T] R is the triangular region with vertices ( 0 , 0 ) , ( 1 , 1 ) , and ( 0 , 5 ) ; ρ ( x , y ) = x + y .

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[T] R is the rectangular region with vertices ( 0 , 0 ) , ( 0 , 3 ) , ( 6 , 3 ) , and ( 6 , 0 ) ; ρ ( x , y ) = x y .

a. M x = 216 2 5 , M y = 432 2 5 ; b. x = 18 5 , y = 9 5 ;
c.
A rectangle R bounded by the x and y axes and the lines x = 6 and y = 3 with point marked (18/5, 9/5).

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[T] R is the rectangular region with vertices ( 0 , 1 ) , ( 0 , 3 ) , ( 3 , 3 ) , and ( 3 , 1 ) ; ρ ( x , y ) = x 2 y .

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[T] R is the trapezoidal region determined by the lines y = 1 4 x + 5 2 , y = 0 , y = 2 , and x = 0 ; ρ ( x , y ) = 3 x y .

a. M x = 368 5 , M y = 1552 5 ; b. x = 92 95 , y = 388 95 ;
c.
A trapezoid R bounded by the x and y axes, the line y = 2, and the line y = negative x/4 + 2.5 with the point marked (92/95, 388/95).

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[T] R is the trapezoidal region determined by the lines y = 0 , y = 1 , y = x , and y = x + 3 ; ρ ( x , y ) = 2 x + y .

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[T] R is the disk of radius 2 centered at ( 1 , 2 ) ; ρ ( x , y ) = x 2 + y 2 2 x 4 y + 5 .

a. M x = 16 π , M y = 8 π ; b. x = 1 , y = 2 ;
c.
A circle with radius 2 centered at (1, 2), which is tangent to the x axis at (1, 0) and has pointed marked at the center (1, 2).

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[T] R is the unit disk; ρ ( x , y ) = 3 x 4 + 6 x 2 y 2 + 3 y 4 .

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[T] R is the region enclosed by the ellipse x 2 + 4 y 2 = 1 ; ρ ( x , y ) = 1 .

a. M x = 0 , M y = 0 ; b. x = 0 , y = 0 ;
c.
An ellipse R with center the origin, major axis 2, and minor axis 0.5, with point marked at the origin.

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[T] R = { ( x , y ) | 9 x 2 + y 2 1 , x 0 , y 0 } ; ρ ( x , y ) = 9 x 2 + y 2 .

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[T] R is the region bounded by y = x , y = x , y = x + 2 , and y = x + 2 ; ρ ( x , y ) = 1 .

a. M x = 2 , M y = 0 ; b. x = 0 , y = 1 ;
c.
A square R with side length square root of 2 rotated 45 degrees, with corners at the origin, (2, 0), (1, 1), and (negative 1, 1). A point is marked at (0, 1).

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[T] R is the region bounded by y = 1 x , y = 2 x , y = 1 , and y = 2 ; ρ ( x , y ) = 4 ( x + y ) .

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In the following exercises, consider a lamina occupying the region R and having the density function ρ given in the first two groups of Exercises.

  1. Find the moments of inertia I x , I y , and I 0 about the x -axis , y -axis , and origin, respectively.
  2. Find the radii of gyration with respect to the x -axis , y -axis , and origin, respectively.

R is the triangular region with vertices ( 0 , 0 ) , ( 0 , 3 ) , and ( 6 , 0 ) ; ρ ( x , y ) = x y .

a. I x = 243 10 , I y = 486 5 , and I 0 = 243 2 ; b. R x = 3 5 5 , R y = 6 5 5 , and R 0 = 3

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R is the triangular region with vertices ( 0 , 0 ) , ( 1 , 1 ) , and ( 0 , 5 ) ; ρ ( x , y ) = x + y .

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R is the rectangular region with vertices ( 0 , 0 ) , ( 0 , 3 ) , ( 6 , 3 ) , and ( 6 , 0 ) ; ρ ( x , y ) = x y .

a. I x = 2592 2 7 , I y = 648 2 7 , and I 0 = 3240 2 7 ; b. R x = 6 21 7 , R y = 3 21 7 , and R 0 = 3 105 7

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R is the rectangular region with vertices ( 0 , 1 ) , ( 0 , 3 ) , ( 3 , 3 ) , and ( 3 , 1 ) ; ρ ( x , y ) = x 2 y .

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R is the trapezoidal region determined by the lines y = 1 4 x + 5 2 , y = 0 , y = 2 , and x = 0 ; ρ ( x , y ) = 3 x y .

a. I x = 88 , I y = 1560 , and I 0 = 1648 ; b. R x = 418 19 , R y = 7410 19 , and R 0 = 2 1957 19

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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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