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Unformatted text preview: Accumulation Authors: Bill Davis, Horacio Porta and Jerry Uhl ©19962007 Publisher: Math Everywhere, Inc. Version 6.0 2.03 Measurements BASICS B.1) Measurements based on slicing and accumulating: Area and volume · B.1.a) Here is the plot of the function f @ x D = 0.7 + 2.1 x + Sin @ 5.3 x D for a § x § b with a = 1 and b = 3 shown with some boxes. Clear @ x, y, f, points, iterations, jump, n, box, boxes, k D ; a = 1; b = 3; f @ x _ D = 0.7 + 2.1 x + Sin @ 5.3 x D ; fplot = Plot @ f @ x D , 8 x, a, b < , PlotStyle> 88 Blue, Thickness @ 0.01 D<< , AxesLabel> 8 "x" , "" <D ; iterations @ n _ D = n; jump @ n _ D = b a iterations @ n D ; points @ n _ D : = Table @ Graphics @8 PointSize @ 0.01 D , Red, Point @8 x, 0 <D<D , 8 x, a, b jump @ n D , jump @ n D<D box @ n _ , x _ D : = 8 Graphics @8 Yellow, Polygon @88 x, 0 < , 8 x, f @ x D< , 8 x + jump @ n D , f @ x D< , 8 x + jump @ n D , 0 <<D<D , Graphics @ Line @88 x, 0 < , 8 x, f @ x D< , 8 x + jump @ n D , f @ x D< , 8 x + jump @ n D , 0 < , 8 x, 0 <<DD< ; boxes @ n _ D : = Table @ box @ n, x D , 8 x, a, b jump @ n D , jump @ n D<D ; AreaStory @ n _ D : = Show @ boxes @ n D , fplot, points @ n D , PlotLabel> n " = n" , AspectRatio> 1 D ; Show @ AreaStory @ 20 DD 20 = n The area measurement Ÿ a b f @ x D „ x is nearly the same as the accumulated measurements of the areas of the boxes. The area of the box whose lower left corner is at the point 8 x, 0 < measures out to f @ x D * jump. So the accumulated area of all the boxes measures out to Sum @ f @ x D jump, 8 x, a, b jump, jump <D As n Ø ¶ , jump Ø 0, these sums close in on Integrate @ f @ x D , 8 x, a, b <D = Ÿ a b f @ x D „ x. See what happens as n gets large and the jump gets small : ListAnimate @8 AreaStory @ 5 D , AreaStory @ 10 D , AreaStory @ 20 D , AreaStory @ 50 D , AreaStory @ 100 D<D 20 = n Reflect on the animation. Then rerun everything with different choices of f @ x D . When the jump between consecutive points is so small that it cannot be measured, a lot of folks (especially engineers) like to say that the jump is „ x. In this case, Sum @ f @ x D jump, 8 x, 1, 3 jump, jump <D = Sum @ f @ x D „ x, 8 x, 1, 3 „ x, „ x <D For all practical purposes, this is the same as Integrate @ f @ x D , 8 x, 1, 3 <D = Ÿ 1 3 f @ x D „ x. So what? · Answer: Look at this: Clear @ bars, jump D ; bars @ jump _ D : = Graphics @8 GrayLevel @ 0.3 D , Table @ Line @88 x, 0 < , 8 x, f @ x D<<D , 8 x, a, b, jump <D<D ; Show B fplot, bars B b a 45 FF 1.5 2.0 2.5 3.0 x 3 4 5 6 7 Put in more bars: Show B fplot, bars B b a 120 FF 1.5 2.0 2.5 3.0 x 3 4 5 6 7 Put in more bars if you like. The upshot is this: If for each x in the plotting interval, you run a bar from 8 x, 0 < to 8 x, f @ x D< and think of it as a box of height f @ x D and of width „ x, then the area under the plotted part of the f @ x D curve and over the interval from x = a to x = b is Sum @ f @ x D „ x, 8 x, a, b „ x, „ x <D and this is Ÿ a b f @ x D...
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This note was uploaded on 10/11/2011 for the course MATH 231 taught by Professor Staff during the Spring '08 term at University of Illinois, Urbana Champaign.
 Spring '08
 Staff

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