Calculus: Single and Multivariable

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Unformatted text preview: Week #10 - The Integral Section 5.3 From “Calculus, Single Variable” by Hughes-Hallett, Gleason, McCallum et. al. Copyright 2005 by John Wiley & Sons, Inc. This material is used by permission of John Wiley & Sons, Inc. SUGGESTED PROBLEMS 1. If f ( t ) is measured in dollars per year and t is measured in years, what are the units of integraldisplay b a f ( t ) dt ? The units are the product of the the function, f ( t ), and dt (or Δ t ): (dollars / year) · (years) = dollars. In Exercises 4-7, explain in words what the integral represents and give units. 5. integraldisplay 6 a ( t ) dt , where a ( t ) is acceleration in km/hr 2 and t is time in hours. As seen in class, the integral of velocity is the net change in position, so we can infer that the integral of acceleration is the net change in velocity. In particular, the net change in velocity under the acceleration a(t) during the time t = 0 to t = 6. The units bear this out: (km/hr 2 ) · (hr) = km/hr. 25. (a) Using Figure 5.42, estimate integraldisplay 3- 3 f ( x ) dx . (b) Which of the following average values of f ( x ) is larger? (i). Between x =- 3 and x = 3 (ii). Between x = 0 and x = 3 Figure 5.42 (a) The integral is the area above the x-axis minus the area below the x- axis. By counting squares, we can estimate that integraldisplay 3- 3 f ( x ) dx is about- 6 + 2 =- 4 (the negative of the area from t =- 3 to t = 1 plus the area from t = 1 to t = 3.) (b) Since the integral in part (a) is negative, the average value of f ( x ) between x =- 3 and x = 3 is negative. From the graph, however, it appears that the average value of f ( x ) from x = 0 to x = 3 is positive. Hence (ii) is the larger quantity. 1 34. A bicyclist pedals along a straight road with velocity, v , given in Figure 5.45. She starts 5 miles from a lake; positive velocities take her away from the lake and negative velocities take her toward the lake. When is the cyclist farthest from the lake, and how far away is she then? Figure 5.45 First, it helps to interpret the graph. Each t-axis square represents 1 / 6th of an hour, or 10 minutes. Each square then represents a distance of (10 miles per hour) · (1 / 6 hour = 5 3 miles. For the first ten minutes, the graph is below the t axis, so her velocity is negative, meaning she is moving towards the lake. At exactly the 20 minute mark, her velocity is zero, so she has stopped....
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