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5.1 |
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 | 1. Suppose the functions f and g satisfy f (x)  g(x) 0 for all x in the interval [a, b]. Explain in terms of the areas  and  why the area between the curves y = f (x) and y = g(x) is given by 
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| | 2. Suppose the functions f and g satisfy f (x)  g(x) 0 for all x in the interval [a, b]. Explain in terms of the areas  and  why the area between the curves y = f (x) and y = g(x) is given by 
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| | 3. Suppose that the speeds of racing cars A and B are vA(t ) and vB(t ) mph, respectively. If vA(t ) vB(t ) for all t, vA(0) = vB(0) and the race lasts from t = 0 to t = 2 hours, explain why car A will win the race by  miles. |
| | 4. Suppose that the speeds of racing cars A and B are vA(t ) and vB(t ) mph, respectively. If vA(t ) vB(t ) for 0 t 0.5 and 1.1 t 1.6 and vB(t ) vA(t ) for 0.5 t 1.1 and 1.6 t 2, describe the difference between  and  Which integral will tell you which car wins the race? |
| In exercises 5-12, find the area between the curves on the given interval.
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| 5. y = x3, y = x2-1, 1 x 3
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| 6. y = cos x, y = x2+2, 0 x 2
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| 7. y = ex, y = x-1, -2 x 0
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| 8. y = e-x, y = x2, 1 x 4
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| 9. y = x2-1, y = 1-x, 0 x 2
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| 10. y = x2-3, y = x-1, 0 x 3
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| 11. y = x3-1, y = 1-x, -2 x 2
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| 12. y = x4+x-2, y = x-1, -2 x 2
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| In exercises 13-20, sketch and find the area of the region determined by the intersections of the curves.
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| 13. y = x2-1, y = 7-x2 | 14. y = x2-1, y =  x2 |
| 15.  | 16. y = x2-x-4, y = x+4 |
| 17. y = x3, y = 3x+2 | 18. y = x3-2x2, y = x2 |
| 19. y = x3, y = x2 | 20. y =  , y = x2
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| In exercises 21-26, sketch and estimate the area determined by the intersections of the curves.
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| 21. y = ex, y = 1-x2 | 22. y = x4, y = 1-x |
| 23. y = sin x, y = x2 | 24. y = cos x, y = x4 |
| 25. y = x4, y = 2+x | 26. y = ln x, y = x2-2
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| In exercises 27-34, sketch and find the area of the region bounded by the given curves. Choose the variable of
integration so that the area is written as a single integral.
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| 27. y = x, y = 2-x, y = 0
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| 28. y = 2x (x > 0), y = 3-x2, x = 0
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| 29. x = 3y, x = 2+y2 | 30. x = y2, x = 1 |
| 31. x = y, x = -y, x = 1 | 32. y = x, y = -x, y = 2
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| 33. y = x, y = 2, y = 6-x, y = 0
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| 34. x = y2, x = 4
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| 35. The average value of a function f (x) on the interval [a, b] is  Compute the average value of f (x) =
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| x2 on [0, 3] and show that the area above y = A and below y = f (x) equals the area below y = A and above the x - axis. |
| 36. Prove that the result of exercise 35 is always true by showing that 
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| 37. The United States oil consumption for the years 1970-1974 was approximately equal to f (t ) = 16.1e0.07t million barrels per day, where t = 0 corresponds to 1970. Following an oil shortage in 1974, the country's consumption changed and was better modeled by g(t ) = 21.3e0.04(t-4) million barrels per day, t 4. Show that f (4) = g(4) and explain what this number represents. Compute the area between f (t ) and g(t ) for 4 t 10. Use this number to estimate the number of barrels of oil saved by Americans' reduced oil consumption from 1974 to 1980.
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| 38. Suppose that a nation's fuelwood consumption is given by 76e0.03t m 3 /yr and new tree growth is 50-6e0.09t m 3 /yr. Compute and interpret the area between the curves for 0 t 10.
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39. Suppose that the birth rate for a certain population is b(t ) = 2e0.04 million people per year, and the death rate for the same population is d(t ) = 2e0.02 million people per year. Show that b(t ) d(t ) for t 0, and explain why the area between the curves represents the increase in population. Compute the increase in population for 0 t 10.
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40. Suppose that the birth rate for a population is b(t ) = 2e0.04t million people per year, and the death rate for the same population is d(t ) = 3e0.02t million people per year. Find the intersection T of the curves ( T > 0 ). Interpret the area between the curves for 0 t T and the area between the curves for T t 30. Compute the net change in population for 0 t 30.
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| 41. In collisions between a ball and a striking object (e.g., baseball bat or tennis racket), the ball changes shape, first compressing and then expanding. If x represents the change in size of the ball (e.g., in inches) for 0 x m and f (x) represents the force between ball and striking object (e.g., in pounds), the area under the curve y = f (x) is proportional to the energy transferred. Suppose that f c(x) is the force during compression and f e(x) is the force during expansion. Explain why  is proportional to the energy lost by the ball (due to friction) and thus   is the proportion of energy lost in the collision. For a baseball and bat, reasonable values are shown (see Adair's book The Physics of Baseball): |
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| x (in.) | 0 | 0.1 | 0.2 | 0.3 | 0.4 |
| f c(x) (lb) | 0 | 250 | 600 | 1200 | 1750 |
| f e(x) (lb) | 0 | 10 | 100 | 270 | 1750 |
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| Use Simpson's rule to estimate the proportion of energy retained by the baseball. |
| 42. Using the same notation as in exercise 41, values for the force f c(x) during compression and force f e(x) during expansion of a golf ball are given by |
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| x (in.) | 0 | 0.045 | 0.09 | 0.135 | 0.18 |
| f c(x) (lb) | 0 | 200 | 500 | 1000 | 1800 |
| f e(x) (lb) | 0 | 125 | 350 | 700 | 1800 |
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| Use Simpson's rule to estimate the proportion of energy retained by the golf ball. |
| 43. Much like the compression and expansion of a ball discussed in exercises 41 and 42, the force exerted by a tendon as a function of its extension determines the loss of energy. Suppose that x is the extension of the tendon, f s(x) is the force during stretching of the tendon and f r(x) is the force during recoil of the tendon. The data given is for a hind leg tendon of a wallaby (see Alexander's book Exploring Biomechanics): |
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| x (mm) | 0 | 0.75 | 1.5 | 2.25 | 3.0 |
| f s(x) (N) | 0 | 110 | 250 | 450 | 700 |
| f r(x) (N) | 0 | 100 | 230 | 410 | 700 |
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| Use Simpson's rule to estimate the proportion of energy returned by the tendon. |
| 44. The arch of a human foot acts like a spring during walking and jumping, storing energy as the foot stretches (i.e., the arch flattens) and returning energy as the foot recoils. In the data, x is the vertical displacement of the arch, f s(x) is the force on the foot during stretching and f r(x) is the force during recoil (see Alexander's book Exploring Biomechanics): |
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| x (mm) | 0 | 2.0 | 4.0 | 6.0 | 8.0 |
| f s(x) (N) | 0 | 300 | 1000 | 1800 | 3500 |
| f r(x) (N) | 0 | 150 | 700 | 1300 | 3500 |
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| Use Simpson's rule to estimate the proportion of energy returned by the arch. |
| 45. The velocities f (t ) and g(t ) of two falling objects are given by f (t ) = -40-32t ft/s and g(t ) = -30-32t ft/s. Assume that the objects start at the same height at time t = 0. Find and interpret the area between the curves for 0 t 10.
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| 46. The velocities of two runners are given by f (t ) = 10 mph and g(t ) = 10-sin t mph. Find and interpret the integrals  and 
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| 47. The velocities of two racing cars A and B are given by f (t ) = 40(1-e-t ) mph and g(t ) = 20t mph, respectively. Estimate (a) the largest lead for car A and (b) the time at which car B catches up. |
 | 48. At this stage, you can compute the area of any “simple” planar region. For a general figure bounded on the left by a function x = l(y), on the right by a function x = r(y), on top by a function y = t (x) and on the bottom by a function y = b(x) , write the area as a sum of integrals. (Hint: Divide the region into subregions which can be written as the integral of r(y)-l(y) or t (x)-b(x).) |
