Steven Strogatz on math, from basic to baffling.

Mathematical signs and symbols are often cryptic, but the best of them offer visual clues to their own meaning. The symbols for zero, one and infinity aptly resemble an empty hole, a single mark and an endless loop: 0, 1, ∞. And the equals sign, =, is formed by two parallel lines because, in the words of its originator, Welsh mathematician Robert Recorde in 1557, “no two things can be more equal.”

In calculus the most recognizable icon is the integral sign:

Its graceful lines are evocative of a musical clef or a violin’s f-hole — a fitting coincidence, given that some of the most enchanting harmonies in mathematics are expressed by integrals. But the real reason that Leibniz chose this symbol is much less poetic. It’s simply a long-necked S, for “summation.”



As for what’s being summed, that depends on context. In astronomy, the gravitational pull of the sun on the earth is described by an integral. It represents the collective effect of all the minuscule forces generated by each solar atom at their varying distances from the earth. In oncology, the growing mass of a solid tumor can be modeled by an integral. So can the cumulative amount of drug administered during the course of a chemotherapy regimen.

Historically, integrals arose first in geometry, in connection with the problem of finding the areas of curved shapes. As we saw two weeks ago, the area of a circle can be viewed as the sum of many thin pie slices. In the limit of infinitely many slices, each of which is infinitesimally thin, those slices could then be cunningly rearranged into a rectangle whose area was much easier to find. That was a typical use of integrals. They’re all about taking something complicated and slicing and dicing it to make it easier to add up.

In a 3-D generalization of this method, Archimedes (and before him, Eudoxus, around 400 B.C.) calculated the volumes of spheres, cones, barrels, prisms and various other solid shapes by re-imagining them as stacks of many wafers or discs, like a salami sliced thin. By computing the volumes of the varying slices, and then ingeniously integrating them — adding them back together — they were able to deduce the volume of the original whole.

Today we still ask budding mathematicians and scientists to sharpen their skills at integration by applying them to these classic geometry problems. They’re some of the hardest exercises we assign, and a lot of students hate them, but there’s no surer way to hone the facility with integrals needed for advanced work in every quantitative discipline from physics to finance.

One such mind-bender concerns the volume of the 3-D region common to two identical cylinders intersecting in a perpendicular fashion, like stovepipes in a kitchen. It takes an unusual gift of imagination to visualize this shape clearly.

So there’s no shame in admitting defeat and looking for a way to make this shape more palpable. To do so, you can resort to a trick my high school calculus teacher used — take a tin can and cut the top off with metal shears to form a cylindrical coring tool. Then core a large Idaho potato or a piece of Styrofoam from two mutually perpendicular directions. Inspect the resulting shape at your leisure.

Lacking both potato and Styrofoam, we have to settle for trying to convey on a flat screen what this curious solid looks like:

Remarkably, it has square cross-sections, even though it was created from round cylinders. It’s a stack of infinitely many layers, each a wafer-thin square, tapering from a large square in the middle to progressively smaller ones, and finally to single points at the top and bottom.

Computer animations now make it possible to reveal the structure of the shape much more easily and vividly.

Still, picturing the shape is merely the first step. What remains is to determine its volume.

Archimedes managed to find it, but only by virtue of his astounding ingenuity. He used a mechanical method based on levers and centers of gravity, in effect weighing the shape in his mind by balancing it against others he already understood. The downside of his approach, besides the prohibitive brilliance it required, was that it applied only to a limited range of shapes.

These conceptual roadblocks stumped the world’s finest mathematicians for the next 19 centuries … until Gregory, Barrow, Newton and Leibniz established what’s now known as the Fundamental Theorem of Calculus in the mid-1600s. The Fundamental Theorem is a powerful link between integrals and the subject of last week’s column, derivatives. It greatly expands the universe of integrals that can be solved, and it reduces their calculation to grunt work. Nowadays computers can be programmed to use it — and so can students. With its help, even the stovepipe problem that was once a world-class challenge now becomes an exercise within common reach. (For the details of Archimedes’s approach as well as the modern one, consult the references in the notes.)

It’s not practical to state the Fundamental Theorem here (though see the notes for an intuitive analogy). Instead I’ll try to convey why it represented such an enormous advance. It allowed mathematicians to forecast a changing world with much greater precision than had ever been possible.

The simplest kind of change can be handled with algebra. When something changes steadily, at a constant rate, algebra works beautifully. This is the domain of “distance equals rate times time.” For example, a car moving at an unchanging speed of 60 miles per hour will surely travel 60 miles in the first hour, and 120 miles by the end of the second hour.

But what about change that proceeds at a varying rate? Such changing change is all around us — in the accelerating descent of a penny dropped from a tall building, in the ebb and flow of the tides, in the elliptical orbits of the planets, in the circadian rhythms within us. And only calculus can cope with the cumulative effects of changes as non-uniform as these.

For nearly two millennia after Archimedes, just one method existed for predicting the net effect of changing change: add up the varying slices, bit by bit. Most of the time it couldn’t be done. The infinite sums were too hard.

The Fundamental Theorem enabled a lot of these problems to be solved — not all of them, but many more than before. It often gave a shortcut for solving integrals, at least for the elementary functions (sums and products of powers, exponentials, logarithms and trig functions) that describe so many of the phenomena in the natural world.

From this perspective, the lasting legacy of integral calculus is a Veg-O-Matic view of the universe. Newton and his successors taught us that nature unfolds in slices. Virtually all the classical laws of physics discovered in the past 300 years turned out to have this character, whether they describe the motions of particles or the flow of heat, electricity, air or water. Together with the governing laws, the conditions in each slice of time or space determine what will happen in adjacent slices.

The implications were profound. For the first time in history, rational prediction became possible… not just one slice at a time, but with the help of the Fundamental Theorem, by leaps and bounds.

So we’re long overdue to update our slogan for integrals — from “It slices, it dices” to “Recalculating. A better route is available.”

NOTES

For more about the ways that integral calculus has been used to help cancer researchers, see:

D. Mackenzie, “Mathematical modeling of cancer,” SIAM News, Vol. 37, January/February 2004.

H.P. Greenspan, “Models for the growth of a solid tumor by diffusion,” Studies in Applied Mathematics, December 1972, p. 317. The region common to two identical circular cylinders whose axes intersect at right angles is known variously as a Steinmetz solid or a bicylinder. Its volume can be calculated straightforwardly but opaquely by modern techniques. An ancient and much simpler solution was known to both Archimedes and Tsu Ch’ung-Chih. It uses nothing more than the method of slicing and a comparison between the areas of a square and a circle. For a marvelously clear exposition, see Martin Gardner’s column:

M. Gardner, “Mathematical games: Some puzzles based on checkerboards,” Scientific American, Vol. 207 (Nov. 1962), p. 164. And for Archimedes and Tsu Ch’ung-Chih, see:

Archimedes, “The Method,” English translation by T. L. Heath (1912), reprinted by (Dover 1953).

T. Kiang, “An old Chinese way of finding the volume of a sphere,” Mathematical Gazette, Vol. 56 (May 1972), pp. 88-91.

Moreton Moore points out that the bicylinder also has applications in architecture: “The Romans and Normans, in using the barrel vault to span their buildings, were familiar with the geometry of intersecting cylinders where two such vaults crossed one another to form a cross vault.” For this, as well as applications to crystallography, see:

M. Moore, “Symmetrical intersections of right circular cylinders,” Mathematical Gazette, Vol. 58 (Oct. 1974), pp. 181-185. For Archimedes’s application of his mechanical method to the problem of finding the volume of the bicylinder, see Proposition 15, p. 48 of T.L. Heath, “The Method of Archimedes, Recently Discovered by Heiberg” (Cosimo Classics, 2007). It’s interesting that Archimedes viewed his mechanical method as a means for discovering theorems rather than proving them. As he put it, “… certain things first became clear to me by a mechanical method, although they had to be demonstrated by geometry afterwards because their investigation by the said method did not furnish an actual demonstration. But it is of course easier, when we have previously acquired, by the method, some knowledge of the questions, to supply the proof than it is to find it without any previous knowledge.”

That last line offers a timeless lesson about problem solving — when you’re trying to prove something, it helps to know it’s true. For a popular account of Archimedes’s work, see:

R. Netz and W. Noel, “The Archimedes Codex: How a Medieval Prayer Book Is Revealing the True Genius of Antiquity’s Greatest Scientist” (Da Capo Press, 2009). Interactive demonstrations of the bicylinder and other problems in integral calculus are available online. You’ll need to download the free Mathematica Player, which will then allow you to explore hundreds of other interactive demonstrations in all parts of mathematics. Mamikon Mnatsakanian at Caltech has produced a series of animations that illustrate the Archimedean spirit and the power of slicing. My favorite is this visualization of a beautiful relationship among the volumes of a sphere and a certain double-cone and cylinder whose height and radius match those of the sphere. He also shows the same thing more physically by draining an imaginary volume of liquid from the cylinder and pouring it into the other two shapes. Similarly elegant mechanical arguments in the service of math are given in:

M. Levi, “The Mathematical Mechanic: Using Physical Reasoning to Solve Problems” (Princeton University Press, 2009). Michael Starbird has created and filmed a fine series of lectures on the basics of calculus. Here’s an analogy that I hope will shed some light on what the Fundamental Theorem is, and why it’s so helpful. (My colleague Charlie Peskin at NYU suggested it.) Imagine a staircase. The total change in height from the top to the bottom is the sum of the rises of all the steps in between. That’s true even if some of them rise more than others, and no matter how many steps there are.

The Fundamental Theorem of Calculus says something similar for functions — if you integrate the derivative of a function from one point to another, you get the net change in the function between the two points. In this analogy, the function is like the elevation of each step compared to ground level. The rises of individual steps are like the derivative. Integrating the derivative is like summing the rises. And the two points are the top and the bottom.

Why is this so helpful? Suppose you’re given an enormous list of numbers to sum, as occurs whenever you’re calculating an integral by slices. If you can somehow manage to find the corresponding staircase — in other words, if you can find an elevation function for which those numbers are the rises — then computing the integral would be a snap. It’s just the top minus the bottom.

That’s the great speed-up made possible by the Fundamental Theorem. And it’s why we torture all beginning calculus students with months of learning how to find elevation functions, technically called “antiderivatives.”

Thanks to Charlie Peskin, for the staircase analogy; Margaret Nelson, for preparing the line drawing; and Paul Ginsparg, Tim Novikoff, Andy Ruina and Carole Schiffman, for their comments and suggestions.

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