5128_Ch07_pp378-433

More documents

Recommendations

Info

Section 7.3 Volumes 403 2. Identify the limits of integration: y runs from 0 to 2. 3. Integrate to find the volume. V 2 0 2 0 2p( radius)( shell height) shell dy 2py4 y 2 dy 8p Now try Exercise 33(a). 4 2 y 0 2 Figure 7.28 The region and the height of a typical shell in Example 5. o x k [–1, 3.5] by [–0.8, 2.2] Figure 7.29 The base of the paperweight in Example 6. The segment perpendicular to the x-axis at x k is the diameter of a semicircle that is perpendicular to the base. 0 2 y x x π y = 2 sin x Figure 7.30 Cross sections perpendicular to the region in Figure 7.29 are semicircular. (Example 6) EXAMPLE 5 Finding Volumes Using Cylindrical Shells The region bounded by the curves y 4 x 2 , y x, and x 0 is revolved about the y-axis to form a solid. Use cylindrical shells to find the volume of the solid. SOLUTION 1. Sketch the region and draw a line segment across it parallel to the y-axis (Figure 7.28). The segment’s length (shell height) is 4 x 2 x. The distance of the segment from the axis of revolution (shell radius) is x. 2. Identify the limits of integration: The x-coordinate of the point of intersection of the curves y 4 x 2 and y x in the first quadrant is about 1.562. So x runs from 0 to 1.562. 3. Integrate to find the volume. Other Cross Sections radius)( height) shell dx 1.562 2p(x)(4 x 2 x) dx 0 13.327 1.562 V 2p ( shell 0 Now try Exercise 35. The method of cross-section slicing can be used to find volumes of a wide variety of unusually shaped solids, so long as the cross sections have areas that we can describe with some formula. Admittedly, it does take a special artistic talent to draw some of these solids, but a crude picture is usually enough to suggest how to set up the integral. EXAMPLE 6 A Mathematician’s Paperweight A mathematician has a paperweight made so that its base is the shape of the region between the x-axis and one arch of the curve y 2 sin x (linear units in inches). Each cross section cut perpendicular to the x-axis (and hence to the xy-plane) is a semicircle whose diameter runs from the x-axis to the curve. (Think of the cross section as a semicircular fin sticking up out of the plane.) Find the volume of the paperweight. SOLUTION The paperweight is not easily drawn, but we know what it looks like. Its base is the region in Figure 7.29, and the cross sections perpendicular to the base are semicircular fins like those in Figure 7.30. The semicircle at each point x has radius 2s in x sin x and area Ax 1 2 2 psin x2 . continued
404 Chapter 7 Applications of Definite Integrals The volume of the paperweight is V p 0 Ax dx p p 2 sin x 2 dx 0 p 2 NINT sin x2 , x, 0,p p 2 1.570796327. Bonaventura Cavalieri (1598—1647) Cavalieri, a student of Galileo, discovered that if two plane regions can be arranged to lie over the same interval of the x-axis in such a way that they have identical vertical cross sections at every point, then the regions have the same area. This theorem and a letter of recommendation from Galileo were enough to win Cavalieri a chair at the University of Bologna in 1629. The solid geometry version in Example 7, which Cavalieri never proved, was named after him by later geometers. The number in parentheses looks like half of p, an observation that can be confirmed analytically, and which we support numerically by dividing by p to get 0.5. The volume of the paperweight is EXAMPLE 7 2 p 2 • p 2 p 2.47 in 4 3 . Cavalieri’s Volume Theorem Now try Exercise 39(a). Cavalieri’s volume theorem says that solids with equal altitudes and identical cross section areas at each height have the same volume (Figure 7.31). This follows immediately from the definition of volume, because the cross section area function Ax and the interval a, b are the same for both solids. b Same volume a Cross sections have the same length at every point in [a, b]. Same cross-section area at every level Figure 7.31 Cavalieri’s volume theorem: These solids have the same volume. You can illustrate this yourself with stacks of coins. (Example 7) a x b Now try Exercise 43.
Page 1 and 2: Chapter7 Applications of Definite I
Page 3 and 4: 380 Chapter 7 Applications of Defin
Page 25: 402 Chapter 7 Applications of Defin

5128_Ch07_pp378-433

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?