CalculusVolume2-SASG-02-06
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OpenStax Calculus Volume 2
Student Answer and Solution Guide
Chapter 2
Applications of Integration
2.6 Moments and Centers of Mass
Section Exercises
For the following exercises, calculate the center of mass for the collection of masses given.
255.
at and at
Answer: 257.
Unit masses at Answer: 259.
at and at Answer: For the following exercises, compute the center of mass 261.
for Answer: 263.
for Answer: 265.
for Answer: 267.
for
OpenStax Calculus Volume 2
Student Answer and Solution Guide
Answer: 269.
for Answer: For the following exercises, compute the center of mass
. Use symmetry to help locate the center of mass whenever possible.
271.
in the triangle with vertices , , and Answer: For the following exercises, use a calculator to draw the region, then compute the center of mass
. Use symmetry to help locate the center of mass whenever possible.
273.
[T] The region bounded by
, , and Answer: 275.
[T] The region between and Answer: 277.
[T] The region bounded by , Answer: 279.
[T] The region bounded by
and in the first quadrant
Answer:
OpenStax Calculus Volume 2
Student Answer and Solution Guide
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OpenStax Calculus Volume 2
Student Answer and Solution Guide
For the following exercises, use the theorem of Pappus to determine the volume of the shape.
281.
Rotating around the
-axis between and Answer: 283.
A general cylinder created by rotating a rectangle with vertices , , and
around the
-axis. Does your answer agree with the volume of a cylinder?
Answer: For the following exercises, use a calculator to draw the region enclosed by the curve. Find the area and the centroid for the given shapes. Use symmetry to help locate the center of mass whenever possible.
285.
[T] Quarter-circle: , , and Answer: 287.
[T] Lens: and Answer: 289.
[T] Half-ring: , , and Answer: 291.
Find the generalized center of mass between , , and
. Then, use the
Pappus theorem to find the volume of the solid generated when revolving around the y
-axis.
Answer: Center of mass: , volume:
OpenStax Calculus Volume 2
Student Answer and Solution Guide
293.
Use the theorem of Pappus to find the volume of a torus (pictured here). Assume that a disk of radius is positioned with the left end of the circle at ,
and is rotated around the y
-axis.
Answer: Volume:
OpenStax Calculus Volume 2
Student Answer and Solution Guide
Student Project
The Grand Canyon Skywalk
1.
Compute the area of each of the three sub-regions. Note that the areas of regions and
should include the areas of the legs only, not the open space between them. Round answers to
the nearest square foot.
Answer: Let denote the area of Then Let denote the area of Then Let denote the area of Then
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OpenStax Calculus Volume 2
Student Answer and Solution Guide
3.
Calculate the center of mass of each of the three sub-regions.
Answer: Let , , and denote the centers of mass of the three regions. We note that all three regions are symmetric with respect to the y
-axis. Thus Additionally, and are symmetric vertically, so To find , we need to find the moment with respect to the x
-axis, . Looking at , we see it is bounded above by and below by
Then Therefore we have In summary, we have
OpenStax Calculus Volume 2
Student Answer and Solution Guide
5.
Assume the visitor center weighs 2,200,000 lb, with a center of mass corresponding to the center of mass of Treating the visitor center as a point mass, recalculate the center of mass of the system. How does the center of mass change?
Answer: The visitors’ center weighs 2.2 million pounds, so its mass, , is given by We know , so we have . Let denote the center of mass of the system. Then, as before, by symmetry, , and we need to find .
Then the mass of the system, , is given by Thus, Then In summary, Notice that the center of mass is now on land, about 10 feet from the edge of the canyon.
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