73 7373 732009-05-04 03:39:17 UTC / rev 24ffb11e86b9+4Discretization4.1 How many babies? 594.2 Simple integrations 614.3 Full width at half maximum 634.4 Stirling’s approximation 654.5 Pendulum period 684.6 Summary and problems 81Discretization, the next technique, is the opposite extreme to calculus. Cal-culus was invented to analyze changing processes such as planetary orbitsor, as a one-dimensional illustration, the distance a ball free falls duringa time t. The simple computation, distance is velocity times time, failsbecause the velocity is not constant. Therefore the fundamental idea of cal-culus: Over short time intervals, the velocity is almost constant, allowingthe simple distance computation to be used.The shorter the intervals, the smaller the error. Discretization turns thisnoble goal on its head: Lump all processes into at most a few fat rectangles.At the cost of larger errors, calculations simplify drastically.4.1 How many babies?74 7474 7460 4.1 How many babies?2009-05-04 03:39:17 UTC / rev 24ffb11e86b9+5004census dataage (yr)106yrThe first example is to estimate the num-ber of babies in the United States. To de-fine the problem, let’s call a child a babyif it is less than two years old. The mostaccurate estimate of their numbers wouldcome from US census data. From the da-ta, make a graph showing the number ofpeople with a given age. Then integratethe curve over the range t = 0 . . . 2 years.Problem 4.1 Dimensions of the vertical axisWhy is the vertical axis labeled in units of people per year rather than in unitssimply of people? Equivalently, why does the vertical axis have dimensions ofT−1?This method has two problems. First, it depends on the huge statisticalresources of the US Census Bureau. A method that requires such a deusex machina is not generalizable or usable on a desert island. Second, evenwith all that data, the method requires integrating a curve with no ana-lytic form, so the integration must be done numerically. That requirementmakes the method specific to this problem. Mathematics, however, is aboutgenerality and patterns. Surely a method exists with potential to transferto other problems?The mention of calculus suggests, to a sufficiently ornery mind, its oppo-site: discretization. Rather than integrating the population curve exactly –a difficult task because of its fluctuations – replace it by a single rectangle.What are the dimensions of this rectangle?The rectangle’s width is a time, and a natural population-related time is lifeexpectancy. So take τ ∼ 80 years as its width. In this discretized model, thepopulation curve is flat in the range t = 0..80 yr, so all people live happily,then die abruptly on their 80th birthday. The height does not have suchan obvious natural value. However, we know the rectangle’s area: It isthe population of the United States, roughly 3 ·108in 2008. Therefore, therectangle’s height is75 7575 754 Discretization 612009-05-04 03:39:17 UTC / rev 24ffb11e86b9+height ∼areawidth∼3 ·10875 yr,Why did the life expectancy change from 80 to 75 years?Fudging the life expectancy simplifies the mental calculations: The newnumber 75 divides into 3 and 300 more easily than 80 does. The result-ing inaccuracy is no worse than in replacing a varying population curvewith a rectangle. With luck, the numerical error may compensate for therectangle-replacement error. With the numerical fudge, the height isheight ∼ 4 ·106yr−1.Area ∼270× 3 · 108∼ 107discretized distribution2 7004census dataAge (years)106yrIntegrating a rectangle ofthat height over the the ranget = 0 . . . 2 yr gives:Nbabies∼ 4 ·106yr−1| {z }height× 2 years| {z }infancy= 8·106.The true number is almostexactly the same as the pre-ceding estimate. As oftenhappens when making ap-proximations, the two errors canceled.Problem 4.2 Landfill volumeEstimate the landfill volume used each year by disposable diapers (nappies).Problem 4.3 CostEstimate the annual revenue of the US diaper industry.4.2 Simple integrations76 7676 7662 4.2 Simple integrations2009-05-04 03:39:17 UTC / rev 24ffb11e86b9+010 1. . .e−ttIn the number-of-babies example, discretiza-tion helped integrate an unknown function(or, rather, a function that required a lot ofwork to determine). Integration is a diffi-cult operation, so discretization can be usefuleven with known functions.Consider the following integral:Z∞0e−tdt.Instead of dividing the area into thin vertical rectangles – the calculusmethod – replace it with one rectangle. For its height, a natural choiceis the maximum height of e−t, namely 1.Its width is harder to choose. If the rectangle is too wide, it overestimatesthe area under the curve, which lies under the rectangle. If is too narrow,it underestimates the area by excluding too large a region from the rectan-gle. In the first case, the curve has fallen too much by the time it escapesthe rectangle; in the second case, the curve has not fallen enough. Thehappy medium is to require that the curve has fallen ‘significantly’ whenit leaves the rectangle. With luck, the overestimate in area from using thecurve’s maximum height as the rectangle’s height will compensate for theunderestimate in neglecting the region outside the rectangle.fake e−t010 1te−tA reasonable criterion for significance is fallingby a factor of 2. This change happens when tincreases by one half-life or ln 2. An alterna-tive criterion is less familiar but it compen-sates with simplicity: falling by a factor of e.With f(t) = e−t, this change happens when tgoes from t to t + 1. The ‘fall by a factor of e’criterion makes the rectangle’s width 1. The resulting rectangle is a unitsquare, and its area exactly matches the integral:Z∞0e−tdt = 1.Problem 4.4Discretize to find77 7777 774 Discretization 632009-05-04 03:39:17 UTC / rev 24ffb11e86b9+Z∞0e−atdt.Check your answer using dimensions and easy cases.Problem 4.5 Cone free-fall distanceFor the falling cones of Section 3.4, the analysis computed only the terminal ve-locity, and the home experiment involved dropping the cones from a height of 1or 2 m. Estimate how far a cone falls before it reaches a significant fraction of itsterminal velocity. Is it a significant fraction of the fall height of 1 or 2 m?4.3 Full width at half maximume−x20 1-1For the Gaussian integralZ∞−∞e−x2dxSection 3.1.2 explained the polar-coordinates trick to show that it is√π.That particular trick works only for an infinite