chapter28 - C H A 28 P T E R More Applications of...

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28 CHAPTER More Applications of Integration 28.1 COMPUTING VOLUMES Volumes by Slicing We compute the signed area of a region in the plane using a divide-and-conquer technique. To Fnd the area under the graph of f we slice the region into n thin slices, each of width 1x = b a n and approximate the area of each slice by the area of a rectangle. Let x i = a + i1x , for i = 0, 1, ... , n . x = ax = b f ( x ) Area f ( x i ) x n i =1 Where we approximate the height of the i th slice by f ( x i ). Figure 28.1 Summing the areas of the slices and taking the limit as the number of slices increases without bound gives us the area in question. area = lim n →∞ n X i = 1 f(x i )1x = Z b a f(x)dx We’ll take a similar approach to calculating volume. Suppose we want to Fnd the volume of a loaf of bread. It could be a plain shape, like a typical loaf of rye bread, or it could be a more complicated shape, like a braided loaf of challah. Whatever the loaf looks like, put the whole thing in a bread slicer and ask for thin slices. The volume of the i th slice, V i , can be approximated by multiplying the area of one of the faces, A(x i ) , by the thickness of the slice, 1x . Suppose we have n slices each of thickness 1x . 853
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854 CHAPTER 28 More Applications of Integration x = ax = b Volume = V i A ( x i ) x n i =1 n i =1 A ( x i ) x x } Figure 28.2 The thinner the slices, the less error is involved in approximating the volume of a slice by the area of a face times the thickness. Let’s attach coordinates to the problem. Set the loaf down along the x -axis and denote the positions of the ends of the loaf by x = a and x = b as shown in Figure 28.2. Slice perpendicular to the x -axis, partitioning [ a , b ] into n equal pieces, each of length 1x = b a n . Let x i = a + i1x ,so a = x 0 <x 1 2 < ··· n = b .We’ll refer to this as a standard partition of [ a , b ]. Let A(x i ) denote the cross-sectional area cut by a plane perpendicular to the x -axis at x i . Then volume = lim n →∞ n X i = 1 A(x i )1x = Z b a A(x) dx . Naturally, this approach can be generalized from a loaf of bread to other solids. u EXAMPLE 28.1 The Egyptians built the Great Pyramid of Cheops in Giza around 2600 b.c . Its original height was about 481 feet and its cross sections are nearly perfect squares. The base is a square whose sides measure about 756 feet. 1 Find its volume. 756 ft Pyramid of Cheops (a) (b) (c) 756 481 y x y x s i x i Figure 28.3 SOLUTION In your mind, pick up this massive pyramid and set it along the x -axis so the x -axis runs along the height of the pyramid and pierces the square base at its center. Start chopping at 1 Facts from David Burton, The History of Mathematics: An Introduction , McGraw-Hill Companies, Inc., 1997, p 56.
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28.1 Computing Volumes 855 x = 0 and stop at x = 481, slicing the interval [0, 481] into n equal subintervals of length 1x . Let x i = i1x .
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chapter28 - C H A 28 P T E R More Applications of...

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