5128_Ch07_pp378-433

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Section 7.5 Applications from Science and Statistics 421 This is a Riemann sum for the function 57p410 yy 2 on the interval from y 0 to y 8. The work of pumping the oil to the rim is the limit of these sums as the norms of the partitions go to zero. 57 p 10 yy 4 2 dy 57 p 10y 4 2 y 3 dy 0 0 57 p 4 [ 10 y 3 y 4 8 3 4 ] 30,561 ft • lb 0 Now try Exercise 17. W 8 8 Figure 7.43 To withstand the increasing pressure, dams are built thicker toward the bottom. Fluid Force and Fluid Pressure We make dams thicker at the bottom than at the top (Figure 7.43) because the pressure against them increases with depth. It is a remarkable fact that the pressure at any point on a dam depends only on how far below the surface the point lies and not on how much water the dam is holding back. In any liquid, the fluid pressure p (force per unit area) at depth h is lb b p wh, Dimensions check: f t2 f lt3 ft, for example where w is the weight-density (weight per unit volume) of the liquid. EXAMPLE 4 The Great Molasses Flood of 1919 Typical Weight-densities (lb/ft 3 ) Gasoline 42 Mercury 849 Milk 64.5 Molasses 100 Seawater 64 Water 62.4 90 ft 1 ft SHADED BAND NOT TO SCALE 90 ft Figure 7.44 The molasses tank of Example 4. At 1:00 P.M. on January 15, 1919 (an unseasonably warm day), a 90-ft-high, 90-footdiameter cylindrical metal tank in which the Puritan Distilling Company stored molasses at the corner of Foster and Commercial streets in Boston’s North End exploded. Molasses flooded the streets 30 feet deep, trapping pedestrians and horses, knocking down buildings, and oozing into homes. It was eventually tracked all over town and even made its way into the suburbs via trolley cars and people’s shoes. It took weeks to clean up. (a) Given that the tank was full of molasses weighing 100 lbft 3 , what was the total force exerted by the molasses on the bottom of the tank at the time it ruptured? (b) What was the total force against the bottom foot-wide band of the tank wall (Figure 7.44)? continued
422 Chapter 7 Applications of Definite Integrals SOLUTION (a) At the bottom of the tank, the molasses exerted a constant pressure of 90 y k 45 Figure 7.45 The 1-ft band at the bottom of the tank wall can be partitioned into thin strips on which the pressure is approximately constant. (Example 4) p wh ( 100 lb ft3) (90 ft) 9000 lb ft2. Since the area of the base was p45 2 , the total force on the base was ( lb 9000 f t2) (2025 p ft2 ) 57,225,526 lb. (b) We partition the band from depth 89 ft to depth 90 ft into narrower bands of width Δy and choose a depth y k in each one. The pressure at this depth y k is p wh 100 y k lbft 2 (Figure 7.45). The force against each narrow band is approximately pressure area 100y k 90p Δy 9000p y k Δy lb. Adding the forces against all the bands in the partition and passing to the limit as the norms go to zero, we arrive at F 90 89 9000pydy 9000p 90 ydy 2,530,553 lb 89 for the force against the bottom foot of tank wall. Now try Exercise 25. 1 12 y 2 Area = 1 4 5 12 Figure 7.46 The probability that the clock has stopped between 2:00 and 5:00 can be represented as an area of 14. The rectangle over the entire interval has area 1. x Normal Probabilities Suppose you find an old clock in the attic. What is the probability that it has stopped somewhere between 2:00 and 5:00? If you imagine time being measured continuously over a 12-hour interval, it is easy to conclude that the answer is 14 (since the interval from 2:00 to 5:00 contains one-fourth of the time), and that is correct. Mathematically, however, the situation is not quite that clear because both the 12-hour interval and the 3-hour interval contain an infinite number of times. In what sense does the ratio of one infinity to another infinity equal 14? The easiest way to resolve that question is to look at area. We represent the total probability of the 12-hour interval as a rectangle of area 1 sitting above the interval (Figure 7.46). Not only does it make perfect sense to say that the rectangle over the time interval 2, 5 has an area that is one-fourth the area of the total rectangle, the area actually equals 14, since the total rectangle has area 1. That is why mathematicians represent probabilities as areas, and that is where definite integrals enter the picture. Improper Integrals More information about improper integrals like fx dx can be found in Section 8.3. (You will not need that information here.) DEFINITION Probability Density Function (pdf) A probability density function is a function f x with domain all reals such that f x 0 for all x and f x dx 1. Then the probability associated with an interval a, b is b f x dx. a Probabilities of events, such as the clock stopping between 2:00 and 5:00, are integrals of an appropriate pdf.
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