Chapter 05 - 5E-05(pp 314-323 3:37 PM Page 314 CHAPTER 5 To...

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Integrals To compute an area we approximate a region by rectangles and let the number of rectangles become large. The precise area is the limit of these sums of areas of rectangles. C H A P T E R 5 5E-05(pp 314-323) 1/17/06 3:37 PM Page 314
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In Chapter 2 we used the tangent and velocity problems to introduce the derivative, which is the central idea in differ- ential calculus. In much the same way, this chapter starts with the area and distance problems and uses them to for- mulate the idea of a definite integral, which is the basic concept of integral calculus. We will see in Chapters 6 and 9 how to use the inte- gral to solve problems concerning volumes, lengths of curves, population predic- tions, cardiac output, forces on a dam, work, consumer surplus, and baseball, among many others. There is a connection between integral calculus and differential calculus. The Fundamental Theorem of Calculus relates the integral to the derivative, and we will see in this chapter that it greatly simplifies the solution of many problems. |||| 5.1 Areas and Distances In this section we discover that in trying to find the area under a curve or the distance traveled by a car, we end up with the same special type of limit. The Area Problem We begin by attempting to solve the area problem: Find the area of the region that lies under the curve from to . This means that , illustrated in Figure 1, is bounded by the graph of a continuous function [where ], the vertical lines and , and the -axis. In trying to solve the area problem we have to ask ourselves: What is the meaning of the word area ? This question is easy to answer for regions with straight sides. For a rect- angle, the area is defined as the product of the length and the width. The area of a triangle is half the base times the height. The area of a polygon is found by dividing it into tri- angles (as in Figure 2) and adding the areas of the triangles. FIGURE 2 h b A= bh A™ A=A¡+A™+A£+A¢ A=l w l w 1 2 FIGURE 1 S= s (x, y) | a¯x¯b, 0¯y¯ƒ d 0 y a b x y=ƒ S x=a x=b x x b x a f x 0 f S b a y f x S 315 |||| Now is a good time to read (or reread) A Preview of Calculus (see page 2). It discusses the unifying ideas of calculus and helps put in perspective where we have been and where we are going. 5E-05(pp 314-323) 1/17/06 3:37 PM Page 315
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316 ❙ ❙ ❙ ❙ CHAPTER 5 INTEGRALS However, it isn’t so easy to find the area of a region with curved sides. We all have an intuitive idea of what the area of a region is. But part of the area problem is to make this intuitive idea precise by giving an exact definition of area. Recall that in defining a tangent we first approximated the slope of the tangent line by slopes of secant lines and then we took the limit of these approximations. We pursue a sim- ilar idea for areas. We first approximate the region by rectangles and then we take the limit of the areas of these rectangles as we increase the number of rectangles. The follow- ing example illustrates the procedure.
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