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Chapter 09

# Chapter 09 - 5E-09(pp 582-591 6:20 PM Page 582 CHAPTER 9...

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Integration enables us to calculate the force exerted by water on a dam. C H A P T E R 9 Further Applications of Integration 5E-09(pp 582-591) 1/17/06 6:20 PM Page 582

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We looked at some applications of integrals in Chapter 6: areas, volumes, work, and average values. Here we explore some of the many other geometric applications of integration—the length of a curve, the area of a surface— as well as quantities of interest in physics, engineering, biology, economics, and statistics. For instance, we will investigate the center of gravity of a plate, the force exerted by water pressure on a dam, the flow of blood from the human heart, and the average time spent on hold during a telephone call. |||| 9.1 Arc Length What do we mean by the length of a curve? We might think of fitting a piece of string to the curve in Figure 1 and then measuring the string against a ruler. But that might be difficult to do with much accuracy if we have a complicated curve. We need a precise definition for the length of an arc of a curve, in the same spirit as the definitions we devel- oped for the concepts of area and volume. If the curve is a polygon, we can easily find its length; we just add the lengths of the line segments that form the polygon. (We can use the distance formula to find the distance between the endpoints of each segment.) We are going to define the length of a general curve by first approximating it by a polygon and then taking a limit as the number of seg- ments of the polygon is increased. This process is familiar for the case of a circle, where the circumference is the limit of lengths of inscribed polygons (see Figure 2). Now suppose that a curve is defined by the equation , where f is continuous and . We obtain a polygonal approximation to by dividing the interval into n subintervals with endpoints and equal width . If , then the point lies on and the polygon with vertices , , . . . , , illustrated in Figure 3, is an approximation to . The length L of is approximately the length of this polygon and the approximation gets better as we let n increase. (See Figure 4, where the arc of the curve between and has been magnified and approximations with succes- sively smaller values of are shown.) Therefore, we define the length of the curve FIGURE 3 FIGURE 4 P i P i-1 P i P i-1 P i-1 P i P i-1 P i y P™ P i-1 P i P n y=ƒ 0 x a x i ¤ x i-1 b C L x P i P i 1 C C P n P 1 P 0 C P i x i , y i y i f x i x x 0 , x 1 , . . . , x n a , b C a x b y f x C 583 FIGURE 1 FIGURE 2 5E-09(pp 582-591) 1/17/06 6:20 PM Page 583
584 ❙❙❙❙ CHAPTER 9 FURTHER APPLICATIONS OF INTEGRATION with equation , , as the limit of the lengths of these inscribed polygons (if the limit exists): Notice that the procedure for defining arc length is very similar to the procedure we used for defining area and volume: We divided the curve into a large number of small parts.

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Chapter 09 - 5E-09(pp 582-591 6:20 PM Page 582 CHAPTER 9...

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