Chapter 6 | Applications of Integration
761
448. Below x 2 + y 2 =1 and above y =1− x
Years after 1920 Value ($)
1
63.90
449. Find the mass of ρ = e − x on a disk centered at the origin with radius 4. 450. Find the center of mass for ρ = tan 2 x on x ∈ ⎛ ⎝ − π 4 , π 4 ⎞ ⎠ . 451. Find the mass and the center of mass of ρ =1 on the region bounded by y = x 5 and y = x . For the following exercises, find the requested arc lengths. 452. The length of x for y =cosh( x ) from x =0 to x =2. 453. The length of y for x =3− y from y =0 to y =4 For the following exercises, find the surface area and volume when the given curves are revolved around the specified axis. 454. The shape created by revolving the region between y =4+ x , y =3− x , x =0, and x =2 rotated around the y -axis. 455. The loudspeaker created by revolving y =1/ x from x =1 to x =4 around the x -axis. For the following exercises, consider the Karun-3 dam in Iran. Its shape can be approximated as an isosceles triangle with height 205 mandwidth 388 m. Assume the current depth of the water is 180 m. The density of water is 1000 kg/m 3 . 456. Find the total force on the wall of the dam. 457. You are a crime scene investigator attempting to determine the time of death of a victim. It is noon and 45°F outside and the temperature of the body is 78°F. You know the cooling constant is k = 0.00824°F/min. When did the victim die, assuming that a human’s temperature is 98°F ? For the following exercise, consider the stock market crash in 1929 in the United States. The table lists the Dow Jones industrial average per year leading up to the crash.
3
100
5
110
7
160
9
381.17
Source : http://stockcharts.com/ freecharts/historical/ djia19201940.html
458. [T] The best-fit exponential curve to these data is givenby y = 40.71 + 1.224 x . Why do you think the gains of the market were unsustainable? Use first and second derivatives to help justify your answer. What would this model predict the Dow Jones industrial average to be in 2014 ? For the following exercises, consider the catenoid, the only solid of revolution that has a minimal surface, or zero mean curvature. A catenoid in nature can be found when stretching soap between two rings. 459. Find the volume of the catenoid y =cosh( x ) from x =−1 to x =1 that is created by rotating this curve around the x -axis, as shown here.
460. Find surface area of the catenoid y =cosh( x ) from x =−1 to x =1 that is created by rotating this curve around the x -axis.
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