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Unformatted text preview: 114 Chapter 6: Applications of the Definite Integral in Geometry, Science and Engineering Summary: This chapter focuses upon using the methods of evaluating definite integrals and applying them in various problems. The first problem considered is that of finding the area between two curves. This extends the idea of finding the area underneath a curve or the total area between a function and an interval. The next application is to find the volumes of various objects or solids. Three basic methods are introduced: the method of slicing, the method of washers (or disks) and the method of shells. The latter two methods may only be applied in the case of a volume of revolution when a curve is revolved around a particular axis. After that, finding the length of a curve or the arc length is discussed. Next the surface area of a solid of revolution is investigated and then the average value of a function. Towards the end of the chapter, two physical applications are discussed: work and force from fluid pressure. At the very end of the chapter, hyperbolic functions such as sinh x and cosh x are defined and studied. OBJECTIVES: After reading and working through this chapter you should be able to do the following: 1. Use definite integrals to find the area between two curves (§6 . 1). 2. Use the method of slicing to find the volume of a solid (§6 . 2). 3. Use the method of disks/washers to find the volume of a solid of revolution (§6 . 2). 4. Use the method of cylindrical shells to find the volume of a solid of revolu tion (§6 . 3). 5. Find the arc length of a plane curve (§6 . 4). 6. Find the surface area of a solid of revolution (§6 . 5). 7. Calculate work done by constant and variable forces over a distance (§6 . 6). 115 116 8. Find the center of gravity and centroid of a two dimensional region or thin lamina (§6 . 7). 9. Calculate the force due to fluid pressure on objects submerged in liquid (§6 . 8). 10. Learn the definitions and properties of the hyperbolic functions (§6 . 9). 6.1 Area Between Two Curves PURPOSE: To use definite integrals to calculate the area be tween two curves. Now that area can be described using definite integrals (from the previous chap area problem (see §5 . 1 , 5 . 4 − 5 . 6) ter), the next step is to use definite integrals to describe the area between two curves. The process is rather straightforward. In the case of finding the area under a single function, this can be thought of as finding the areas of many infinites mal rectangles between the function and the axis. Now, these rectangles will be between the two curves. Then the general area function can be stated as integraldisplay b a ( f 1 ( x ) − f 2 ( x )) dx where f 1 ( x ) > f 2 ( x ) ....
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This note was uploaded on 06/01/2010 for the course CAL 1000 taught by Professor Lee during the Spring '10 term at École Normale Supérieure.
 Spring '10
 lee

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