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2.7E: Exercises for Section 6.6

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In exercises 1 - 6, calculate the center of mass for the collection of masses given.

1) m1=2 at x1=1 and m2=4 at x2=2

2) m1=1 at x1=1 and m2=3 at x2=2

Answer
x=54

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

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

Answer
(23,23)

5) m1=1 at (1,0) and m2=4 at (0,1)

6) m1=1 at (1,0) and m2=3 at (2,2)

Answer
(74,32)

In exercises 7 - 16, compute the center of mass ˉx.

7) ρ=1 for x(1,3)

8) ρ=x2 for x(0,L)

Answer
3L4

9) ρ=1 for x(0,1) and ρ=2 for x(1,2)

10) ρ=sinx for x(0,π)

Answer
π2

11) ρ=cosx for x(0,π2)

12) ρ=ex for x(0,2)

Answer
e2+1e21

13) ρ=x3+xex for x(0,1)

14) ρ=xsinx for x(0,π)

Answer
π24π

15) ρ=x for x(1,4)

16) ρ=lnx for x(1,e)

Answer
14(1+e2)

In exercises 17 - 19, compute the center of mass (ˉx,ˉy). Use symmetry to help locate the center of mass whenever possible.

17) ρ=7 in the square 0x1,0y1

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

Answer
(a3,b3)

19) ρ=2 for the region bounded by y=cos(x),y=cos(x),x=π2, and x=π2

In exercises 20 - 26, use a calculator to draw the region, then compute the center of mass (ˉx,ˉy). Use symmetry to help locate the center of mass whenever possible.

20) [T] The region bounded by y=cos(2x),x=π4, and x=π4

Answer
(0,π8)

21) [T] The region between y=2x2,y=0,x=0, and x=1

22) [T] The region between y=54x2 and y=5

Answer
(0,3)

23) [T] Region between y=x,y=lnx,x=1, and x=4

24) [T] The region bounded by y=0 and x24+y29=1

Answer
(0,4π)

25) [T] The region bounded by y=0,x=0, and x24+y29=1

26) [T] The region bounded by y=x2 and y=x4 in the first quadrant

Answer
(58,13)

In exercises 27 - 31, use the theorem of Pappus to determine the volume of the shape.

27) Rotating y=mx around the x-axis between x=0 and x=1

28) Rotating y=mx around the y-axis between x=0 and x=1

Answer
V=mπ3 units³

29) 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?

30) 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?

Answer
V=πa2b units³

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

In exercises 32 - 36, use a calculator to draw the region enclosed by the curve. Find the area M and the centroid (ˉx,ˉy) for the given shapes. Use symmetry to help locate the center of mass whenever possible.

32) [T] Quarter-circle: y=1x2,y=0, and x=0

Answer
(43π,43π)

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

34) [T] Lens: y=x2 and y=x

Answer
(12,25)

35) [T] Ring: y2+x2=1 and y2+x2=4

36) [T] Half-ring: y2+x2=1,y2+x2=4, and y=0

Answer
(0,289π)

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

38) Find the generalized center of mass between y=a2x2,x=0, and y=0. Then, use the Pappus theorem to find the volume of the solid generated when revolving around the y-axis.

Answer
Center of mass: (a6,4a25),
Volume: 2πa49 units³

39) Find the generalized center of mass between y=bsin(ax),x=0, and x=πa. Then, use the Pappus theorem to find the volume of the solid generated when revolving around the y-axis.

40) 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.

Answer
Volume: V=2π2a2(b+a)

41) Find the center of mass (ˉx,ˉy) for a thin wire along the semicircle y=1x2 with unit mass. (Hint: Use the theorem of Pappus.)

Contributors

Gilbert Strang (MIT) and Edwin “Jed” Herman (Harvey Mudd) with many contributing authors. This content by OpenStax is licensed with a CC-BY-SA-NC 4.0 license. Download for free at http://cnx.org.


This page titled 2.7E: Exercises for Section 6.6 is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by OpenStax via source content that was edited to the style and standards of the LibreTexts platform.

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