The Fundamentals of
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In the calculus mathematicians have given us a means to use our discrete mathematics to create accurate descriptions of continuous entities. Because our numbers do not allow us to represent Reality with perfect precision, we resort to a kind of mathematical Impressionism, in particular one like the Pointillism of Georges Seurat as expressed in his famous painting "A Sunday Afternoon on the Island of La Grande Jatte". Like Seurat, we make our pointillist picture as accurate to the scene as we can and then we shrink the pixels until we can no longer discern them: the picture then becomes indistinguishable from the Reality it depicts.
A curve comprises a one-dimensional continuum in a two-dimensional continuum. When we use Descartes' method to plot an algebraic formula, such as y = f(x), as a curve on a plane, we actually play a sophisticated connect-the-dots game. We do not actually draw the continuous curve itself, but, rather, a series of dots that lie on the curve itself. As we draw ever more of those dots, calculating their locations from the formula and them measuring the coordinates that we need to place them correctly on the grid that we have drawn on our plane, we eventually decrease the distance between neighboring dots below the ability of the human eye to resolve the gaps. In that manner we create the illusion of a continuous line without actually drawing a continuum. In that description we see a hint of the fundamental concept of the calculus - that of the limit.
We want a limiting process that describes the process of integration. I say that the integral of the integrand f(x)dx equals the limit as dx approaches zero of the sum of f(x)dx for all values of x in a given range with x = kdx, in which k represents some extremely large number. That does not seem immediately promising, but in the next essay I will show you an elegant calculation of one of the most fundamental forms of the integral done without any reference to differentiation. Integration, you will see, can stand on its own foundation.
In integration, then, we seek the limit of a Riemann sum:
Please note that in this process dx only approaches zero but does not become zero. Indeed, dx does not even become an infinitesimal, but, rather, becomes an extremely small number that we could represent with the inverse of a very large, but nonetheless finite, number. We have a compelling reason for making that choice. If dx becomes equal to zero, then it cannot add up to any non-zero number, even if we multiply it by an infinity, so we could not use it in calculations meant to describe a finite phenomenon. If dx becomes an infinitesimal, the inverse of an infinity, we cannot use it in calculations that we mean to yield consistent answers: the arithmetic of infinity precludes such a thing.
If we call the integral of Equation 1 F(x), then f(x) equals the derivative of F(x) with respect to x. Integration and differentiation are each the inverse of the other, just as addition and subtraction are each the inverse of the other and multiplication and division are each the inverse of the other. Integrals are often described as anti-derivatives, because mathematicians obtain them by looking at examples of derivatives to see what function yields a given derivative: that function is then identified as the integral of the derivative. In these essays, though, I want to avoid anti-differentiation in so far as it is possible: I want to obtain the integrals by actual integration. In order to achieve that goal I need to accumulate enough facts that I can parlay them into the array of integrals that fill the mathematical tables.
Because we define the integral as a sum of terms, we know several facts about the integral right away. We know, for example, that it conforms to the associative and commutative laws of addition, so if we have an integrand of the form (f(x) + g(x))dx, then we have
If we have an integrand of the form Af(x)dx, in which A represents some constant, then we can carry out the addition and the multiplication in either order, so we have
We also have as a consequence of defining the integral as the limit of a Riemann sum:
1. The Most Basic Integral:
We want to integrate the minuscule area adx; that is, we want to add up a series of n rectangles, each covering an area adx, over a distance x = ndx. We have here a rectangle of height a and length ndx = x, so we need not bother with applying the limiting process to the Riemann sum. We have directly
2. The Chain Rule:
Let's imagine that we want to calculate the area of the projection onto the y-z plane of the area under a curve in the z-direction; that is, we want the area of the shadow that the area between the curve and the x-y plane casts onto the y-z plane. We have the description of the curve as z = f(y), but we can only measure in the x-direction. Fortunately we also know that the curve's shadow on the x-y plane conforms to y = g(x), so instead of integrating f(y)dy we will integrate f(g(x))(dg/dx)dx.
3. Integration by Parts:
For this derivation we have an area defined as a rectangle whose borders consist of the ordinate, the abscissa, the line y = f(t), and the line x = g(t). We want to know the area covered by the rectangle at any given time t. We know first of all that we have the area as A = A0 + f(t)g(t), in which A0 represents the area that the rectangle covers at the arbitrary time t = 0. But we also know that we can calculate the area via integration. In the interval dt the rectangle gains or loses area in the amount
If we perform the Riemann summation of that expression with respect to time, we get
The third term on the right drops out because we can feign calculating the value of its Riemann sum. The term represents the area of little rectangles that we must add up as we add up the other contributions to the area of our large rectangle, so we will have n of these little areas to add together. We thus have a term proportional to ndt2 = tdt. The limit to which the term tends as dt tends toward zero equals zero, so the term drops out. Thus we now have the area coming from the sum of two Riemann sums:
That statement gives us the basis for the process of integration by parts.
These comprise the rules of integration that come directly from the definition of an integral as the limit of a Riemann sum. Now I have one final aspect of integration that I must present before we proceed to the integrals themselves.
In order to present integration as the limit of a Riemann sum I have described it as a means of calculating the area under a curve. But now I want to shift metaphors. We can use integration with any process of gradual accumulation, so now I want to calculate the altitude that a train reaches as it climbs a mountain railroad.
We assume that we have a formula that describes the grade of the track as a function of the horizontal distance the train has come from a station at the foot of the mountains. Conventionally, the grade tells us the amount of altitude that the track gains for every one hundred feet of horizontal distance along the track. If we represent that horizontal distance by the coordinate x, then we can express the grade as g(x). Between the points x1 and x2, then, we calculate the change in altitude h(x) as
Suppose that we have the grade function but not the values of x1 and x2. We can still work out the integral, though in this instance we obtain what we call an indefinite integral. We get a formula that looks like it will give us an altitude if we substitute x into it and carry out the indicated arithmetic, but when we carry out the actual calculation we discover that we get the wrong answer. However, if we carry out the calculation for several different values of x, we discover that our answers are all wrong by the same amount. We need only add that amount to our formula to correct it. Indeed, we must make that addition a part of the process of integration, so the indefinite integral that corresponds to Equation 8 is
in which equation C represents the constant of integration.
In the integration constant the indefinite integral gives us a little puzzle to solve. When we want to calculate the altitude of some point with coordinate x from Equation 9, we must ask "altitude relative to what?" Do we determine the altitude relative to that little station at the foot of the mountains? Relative to mean sea level? Relative to some other fiducial mark? Once we answer that question we can determine the value of C and thus complete Equation 9.
Of course, C must be a constant; that is, it must have the same value for all values of x. If we were to pick two points x1 and x2 and use them to calculate the value of h(x2) - h(x1), we would have to obtain Equation 8, turning the indefinite integral into a definite integral. But that only happens if the integration constants in the two indefinite integrals cancel each other out.
Consider another example that demonstrates the use of the integration constant: suppose that the caboose comes uncoupled from the end of a freight train and that its brakes fail to set. The caboose accelerates down the mountain, miraculously remaining on the track, and rolls out onto the prairie at high speed. Let's assume that we know the acceleration as a function of elapsed time (and if the acceleration is uniform, that's a fairly easy criterion to meet). We can then calculate the caboose's speed at any given time by way of
In this case C1 represents the initial speed of the caboose at the time t=0. If we make the replacement C1 = v(0), then we can calculate the caboose's position at any time by integrating Equation 10 with respect to time to get
In this equation the integration constant, C2 = x(0), represents the position of the caboose at the time t=0.
So now we have in mind all of the basic rules that we need to carry out integrations. In the following essays I will show you how to integrate a wide variety of functions and I will show you applications to go with those derivations. I will present the results as indefinite integrals and, in keeping with the convention used in the tables, I will not show the integration constant.
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