2.7E: Exercises for Section 6.6
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- Jun 25, 2021
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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+1e2−1
13) ρ=x3+xe−x for x∈(0,1)
14) ρ=xsinx for x∈(0,π)
- Answer
- π2−4π
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 0≤x≤1,0≤y≤1
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=√1−x2,y=0, and x=0
- Answer
- (43π,43π)
33) [T] Triangle: y=x,y=2−x, 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=a2−x2,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.
- Answer
- Volume: V=2π2a2(b+a)
41) Find the center of mass (ˉx,ˉy) for a thin wire along the semicircle y=√1−x2 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.