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Chapter 6 | Applications of Integration
separately. The first region, denoted R 1 , consists of the curved part of the U. We model R 1 as a semicircular annulus, with inner radius 25 ft and outer radius 35 ft, centered at the origin (see the following figure).
Figure 6.73 We model the Skywalk with three sub-regions.
The legs of the platform, extending 35 ft between R 1 and the canyon wall, comprise the second sub-region, R 2 . Last, the ends of the legs, which extend 48 ft under the visitor center, comprise the third sub-region, R 3 . Assume the density of the lamina is constant and assume the total weight of the platform is 1,200,000 lb (not including the weight of the visitor center; we will consider that later). Use g =32ft/sec 2 . 1. Compute the area of each of the three sub-regions. Note that the areas of regions R 2 and R 3 should include the areas of the legs only, not the open space between them. Round answers to the nearest square foot. 2. Determine the mass associated with each of the three sub-regions. 3. Calculate the center of mass of each of the three sub-regions. 4. Now, treat each of the three sub-regions as a point mass located at the center of mass of the corresponding sub-region. Using this representation, calculate the center of mass of the entire platform. 5. Assume the visitor center weighs 2,200,000 lb, with a center of mass corresponding to the center of mass of R 3 . Treating the visitor center as a point mass, recalculate the center of mass of the system. How does the center of mass change? 6. Although the Skywalk was built to limit the number of people on the observation platform to 120, the platform is capable of supporting up to 800 people weighing 200 lb each. If all 800 people were allowed on the platform, and all of them went to the farthest end of the platform, how would the center of gravity of the system be affected? (Include the visitor center in the calculations and represent the people by a point mass located at the farthest edge of the platform, 70 ft from the canyon wall.)
Theorem of Pappus This section ends with a discussion of the theorem of Pappus for volume , which allows us to find the volume of particular
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