2.3.10. Remarks on Applications to the Natural Sciences:. In applications of mathematics to natural phenomena, we never have to deal with sharply defined quantities. Whether a length is exactly a metre is a question which cannot be decided by any experiment and which consequently has no physical meaning. Again, there is no immediate physical meaning in saying that the length of a material rod is rational or irrational; we can always measure it with any desired degree of accuracy in rational numbers, and the real matter of interest is whether or not we can manage to perform such a measurement using rational numbers with relatively small denominators. Just as the question of rationality or irrationality in the rigorous sense of exact mathematics has no physical meaning, so the actual carrying out of limiting processes in applications will usually be nothing more than a mathematical idealization.

The practical significance of such idealizations lies chiefly in the fact that, if they are used, all analytical expressions become essentially simpler and more manageable. For example, it is vastly simpler and more convenient to work with the notion of instantaneous velocity, which is a function of only one definite instant of time, than with the notion of average velocity between two different instants. Without such an idealization, every rational investigation of nature would be condemned to hopeless complications and would break down at the very outset.

However, we do not intend to enter into a discussion of the relationship of mathematics to reality. We merely wish to emphasize, for the sake of our better understanding of the theory, that we have in applications the right to replace a derivative by a difference quotient and vice versa, provided only that the differences are small enough to guarantee a sufficiently close approximation. The physicist, the biologist, the engineer, or anyone else who has to deal with these ideas in practice, will therefore have the right to identify the difference quotient with the derivative within his limits of accuracy. The smaller is the increment h = dx of the independent variable, the more accurately can it represent the increment Dy = f(x + h) - f(x) by the differential dy = hf'(x). As long as one keeps within the limits of accuracy required by a given problem, one is accustomed to speak of the quantities dx = h and dy = hf '(x) as infinitesimals. These physically infinitesimal quantities have a precise meaning. They are finite quantities, not equal to zero, which are chosen small enough for a given investigation, e.g., smaller than a fractional part of a wave-length or smaller than the distance between two electrons in an atom; in general, they are chosen smaller than the degree of accuracy required.

1.* Replace the statement: At the point x = x , the function f(x) is not differentiable by an equivalent statement without use of any form of the word differentiable.

2. Differentiate the following functions directly by using the definition of the derivatives:

3. Find the intermediate value x of the mean value theorem for the following functions and illustrate graphically:

4. Show that the mean value theorem fails for the following functions when the two points are taken with opposite signs, e.g., x1 = -1, x2 = 1:

Illustrate graphically and compare with the previous example.

As we have already mentioned above, the connection between the problems of integration and of differentiation is the corner-stone of the differential and integral calculus. We will now study this connection.

2.4.1 The Integral as a Function of the Upper Limit: The value of the definite integral of a function f(x) depends on the choice of the two limits of integration a and b; it is a function of the lower limit a as well as of the upper limit b. In order to study this dependence more closely, we imagine the lower limit a to be a definite fixed number, denote the variable of integration no longer by x but by u (Integration of a Linear Function) and the upper limit by x instead of b, in order to suggest that we shall let the upper limit vary and wish to investigate the value of the integral as a function of the upper limit. Accordingly, we write

We call this function F(x) an indefinite integral of the function f(x). When we speak of an and not of the indefinite integral, we suggest that instead of the lower limit a any other could be chosen, in which case we should ordinarily obtain a different value for the integral. Geometrically speaking, the indefinite integral for each value of x will be given by the area (shaded in Fig.17) under the curve y=f(u) and bounded by the ordinates u = a and u = x, the sign being determined by the rules given earlier.

If we choose another lower limit a in place of the lower limit a, we obtain the indefinite integral

The difference Y(x) - F(x) will obviously be

which is a constant, since a and a are each fixed given numbers. Hence

Different indefinite integrals of the same function differ only by an additive constant.

We may likewise regard the integral as a function of the lower limit and introduce the function

in which b is a fixed number. Here again two such integrals with different upper limits b and b differ only by an additive constant

2.4.2 The Derivative of the Indefinite Integral: We will now differentiate the indefinite integral F(x) with respect to the variable x. The result is the theorem:

The indefinite integral

of a continuous function f(x) always possesses a derivative F '(x) and moreover

that is, differentiation of the indefinite integral of a given continuous function always gives us back that function.

This is the basic idea of all of the differential and integral calculus. The proof follows extremely simply from the interpretation of the integral as an area. We form the difference quotient

and observe that the numerator

is the area between the ordinate corresponding to x and the ordinate corresponding to x + h.

Now let x0 be a point in the interval between x and x + h, at which the function f(x) takes its greatest value, and x1 a point, at which it takes its least value in the interval (Fig. 18). Then the area in question will lie between the values hf(x0) and hf(x1) which represent the areas of rectangles with the interval from x to x+h as base and the altitudes f(x0) and f(x1), respectively. Expressed analytically,

This can also be proved directly from the definition of the integral without appealing to the geometrical interpretation (Compare also with the discussion in 2.7). For this purpose, we write

where u0 = x, u1, u2, ··· , un = x + h are points of sub-division of the interval from x to x + h, and the largest of the absolute values of the differences Dun=un - un-1 tends to zero as n increases. Then Dun/h is certainly positive, no matter whether h is positive or negative. Since we know that

and the sum of the quantities Dun is equal to h, it follows that

thus, if we let n tend to infinity, we obtain the above inequalities for

Now, if h tends to zero, both f(x0) and f(x1) must tend to the limit f(x) due to the continuity of the function, whence we see at once that

as stated by the theorem.

Owing to the differentiability of F(x) and 2.3.5, we have the theorem:

The integral of a continuous function f(x) is itself a continuous function of the upper limit.

For the sake of completeness, we point out that, if we regard the definite integral not as a function of its upper limit but as a function of its lower limit, the derivative is not equal to f(x), but instead equal to - f(x); in symbols, if we set

then

The proof follows immediately from the remark that

2.4.3 The Primitive Function; General Definition of the Indefinite Integral: The theorem, which we have just proved, shows that the indefinite integral F(x) at once yields the solution of the problem: Given a function f(x), find a function F(x) such that

This problem requires us to reverse the process of differentiation. It is a typical inverse problem such as occurs in many parts of mathematics and such as we have already found to be a fruitful mathematical tool for generating new functions. (For example, the first extension of the idea of natural numbers was made under the pressure of the necessity for reversing certain elementary computational processes. The formation of inverse functions has led and will lead us to new kinds of functions!)

A function F(x) such that F'(x) = f(x) is called a primitive function of f(x), or simply a primitive of f(x); this terminology suggests that the function f(x) arises from F(x) by differentiation.

This problem of the inversion of differentiation or of the determination of a primitive function is at first sight of quite a different character from the problem of integration. However, we know from the preceding section that:

Every indefinite integral F(x) of the function f(x) is a primitive of f(x).

However, this result does not completely solve the problem of finding primitive functions, because we do not yet know whether we have found all the solutions of the problem. The question concerning the group of all primitive functions is answered by the following theorem, sometimes referred to as the fundamental theorem of the differential and integral calculus:

The difference between two primitives F1(x) and F2(x) of the same function f(x) is always a constant:

Thus, from any one primitive function F(x), we can obtain all the others in the form

by a suitable choice of the constant c. Conversely, for every value of the constant c the expression F1(x) = F(x) + c represents a primitive function of f(x).

It is clear that for any value of the constant c the function F(x) + c is a primitive, provided that F(x) itself is one. From 2.3.4, we have

and since, by assumption, the right-hand side tends to f(x) as h ® 0, so does the left-hand side, whence

Thus, in order to complete the proof of the theorem, there only remains to show that the difference of two primitive functions F1(x) and F2(x) is always a constant. For this purpose, we consider the difference

and form the derivative

By assumption, both the expressions on the right-hand side have the same limit f(x) as h ® 0; thus,for every value of x, we have G'(n) = 0. But a function, the derivative of which is everywhere zero, must have a graph the tangent of which is everywhere parallel to the x-axis, i.e., it must be a constant, whence G(x) = c, as has been stated above. We can prove this last fact by using the mean value theorem without relying on tuition. Applying the mean value theorem to G(x), we find

However, we have seen that the derivative G'(x) is equal to 0 for every value of x, whence, in particular, it is true for the value x, and it follows immediately that G(x1) = G(x2). Since x1 and x2 are arbitrary values of x in the given interval, G(x) must be a constant.

Combining the theorem which has just been proved with the result of 2.4.2, we can now state:

Every primitive function F(x) of a given function f(x) can be represented in the form

where c and a are constants, and conversely, for any arbitrarily chosen constant values a and c, this expression always represents a primitive function.

It may readily be guessed that, as a rule, the constant c can be omitted, since, by changing the lower limit a, we change the primitive function by an additive constant. However, in many cases, we should not obtain all the primitive functions, if we were to omit c, as is shown by the example f(x) = 0. For this function, the indefinite integral of 2.4.1 is always 0, independently of the lower limit; yet any arbitrary constant is a primitive function of f(x) = 0. A second example is the function which is only defined for non-negative values of x. The indefinite integral is

and we see that, independently of the choice of the lower limit a, the indefinite integral of F(x) is always obtained from 2x3/2/3 by addition of a constant which is less than or equal to zero, namely, by the constant -2a3/2; yet such a function as 2x3/2 + 1 is also a primitive of . Thus, in the general expression for the primitive function, we cannot dispense with the additive constant. The relationship which we have found suggests an extension of the idea of the indefinite integral. We shall henceforth call every expression of the form

an indefinite integral of f(x). In other words, we shall no longer make any distinction between the primitive function and the indefinite integral. Nevertheless, if the reader is to have a proper understanding of the interrelationships of these concepts, it is absolutely necessary that he should clearly bear in mind that, in the first instance, integration and inversion of differentiation are two entirely different things, and also that it is only the knowledge of the relationship between them which allows us to apply the term indefinite integral also to the primitive function.

It is customary to represent the indefinite integral by a notation which in itself is perhaps not perfectly clear. We write

that is, we omit the upper limit x and the lower limit a and also the additive constant c and use the letter x for the integration variable. It would really be more consistent to avoid this last change, in order to prevent confusion with the upper limit x which is the independent variable in F(x). When using the notation , we must never lose sight of the indeterminacy connected with it, i.e., the fact that that symbol always denotes only an indefinite integral.

2.4.4 The use of the Primitive Function in the Evaluation of Definite Integrals:. Suppose that we know any one primitive function for the function f(x) and that we wish to evaluate the definite integral We know that the indefinite integral

being also a primitive of f(x), can only differ from F(x) by an additive constant. Consequently,

and the additive constant c is at once determined, if we recollect that the indefinite integral must take the value 0 when x = a. We thus obtain

whence c = - F(a) and F(x)=F(x) - F(a). In particular, we have for the value x=b

which yields the important rule:

If F(x) is any primitive of the function f(x), the definite integral of f(x) between the limits a and b is equal to the difference F(b) - F(a).

If we use the relation F '(x) = f(x), this may be written in the form

This formula can easily be proved and understood directly. We subdivide the interval a £ x £ b into intervals Dx1 Dx2, ··· , Dxn and consider the sum S(DF/Dxn)Dxn. On the one hand, this sum is simply SDF = F(b) - F(a), independently of the particular sub-division, whence its limit is F(b) - F(a). On the other hand, its limit is also equal to as follows from the mean value theorem. In fact, DF/Dxn = F '(xn), where xn is a point between the ends xn-1 and xn, of the interval Dx. The sum is therefore equal to SDxnF '(xn) and, by the definition of the integral, this tends to the limit as the subdivision is made finer, which establishes the theorem.

In applying our rule, we often employ the symbol | to denote the difference F(b) - F(a), i.e., we write

indicating by the vertical line that in the preceding expression first the value b and then the value a is to be substituted for x and finally the difference of the resulting numbers found.

2.4.5 Examples:

We are now in a position to illustrate by a series of simple examples the relationship between the definite integral, the indefinite integral and the derivative, which we have just investigated. By virtue of the theorem of 2.4.2 , we can derive from each of the integration formulae, which have been proved directly in 2.1.2, a differentiation formula.

In 2.2.4, we have obtained the integration formula

for every rational number a ¹ 1 and all positive values of a and b; if we replace the variable of integration by u and the upper limit by x, this may be written in the form

It follows from this by the fundamental theorem that the right hand side is a primitive function of the integrand, i.e., the differentiation formula

is valid for every rational value of a ¹ -1 and all positive values of x. By direct substitution, we find that this last formula is also true for a = -1, if x > 0. The result obtained exactly agrees with what we have already found in 2.3.3 by direct differentiation. Thus, by using the fundamental theorem after having carried out the integration, we could have saved ourselves the trouble of that differentiation.

Moreover, it follows from the integration formula

given in 2.2.5 that d/dx sin x = cos x, in agreement with the result found at the end of 2.3.3.

However, conversely, we may regard every directly proved differentiation formula F'(x) = f(x) as a link between a primitive function F(x) and a derived function f(x), that is, we may regard it as a formula for indefinite integration and then obtain from it the definite integral of f(x) as in 2.4.4. This very method is frequently employed, as we shall see in 4.1. In particular, we may start from the results of 2.3.3 and obtain the integral formula of 2.1.3 by virtue of the fundamental theorem. For example, we know from 2.3.3 that dxa+1/dx = (a+1)xa, whence xa+1/(a+1) is a primitive function or indefinite integral of xa, provided that a ¹ -1, and thus, by 2.4.4 arrive at the above integration formula.

Exercises 2.3:

1. From the differentiations performed in Examples 2, 3 above, set up the corresponding integrations.

2. Evaluate

3. Using Example 2, prove from the definition of the definite integral that

2.5 Simple Methods of Graphical Integration

Since an indefinite integral or primitive function of f(x) is a function y = F(x), which not only can be visualized as an area, but, like any other function, can be represented graphically by a curve. Our definition immediately suggests the possibility of constructing this curve approximately and thus obtaining a graph of the integral function. To begin with, we must remember that this last curve is not unique, but on account of the additive constant can be shifted parallel to itself in the direction of the y-axis. We can therefore require that the integral curve shall pass through an arbitrarily selected point, e.g., if x = l belongs to the interval of definition of f(x), through the point with the coordinates x = 1, y = 0. The curve is thereafter determined by the requirement that for each value of x its direction is given by the corresponding value of f(x). In order to obtain an approximate construction of a curve which satisfies these conditions, we seek to construct not the curve y = F(x) itself, but a polygonal path (broken line) the corners of which lie vertically above previously assigned points of division of the x-axis and the segments of which have approximately the same direction as the portion of the integral curve between the same points of sub-division. For this purpose, we subdivide our interval of the x-axis by means of the points x = 1, x1, x2, ··· into a certain number of parts, not necessarily all of the same length,

and at each point of subdivision we draw a parallel to the y-axis (Fig 19 above). We then draw through the point x= 1, y =0 the straight line the slope of which is equal to f(l); through the intersection of this line with the line x = x1, we draw the line with the slope f(x1); through the intersection of this line with x = x2, we draw the line with the slope f(x2), and so on. In the actual construction of these lines, we erect at each point of sub-division the ordinate to the curve y=f(x) and project these ordinates onto any parallel to the y-axis; in order to be specific, let us suppose that they are projected onto the y-axis itself. We then obtain the direction of the integral curve by joining the points with coordinates x = 0 and y=f(x) to the point x = -1, y = 0. By transferring these directions parallel to themselves, we obtain a polygonal path the corners of which lie vertically above the given points of sub-division of the x-axis and the directions of which agrees with the direction of the integral curve at the starting point of each interval. This polygonal path can be made to represent the integral curve with any desired degree of accuracy by making the subdivisions of the interval fine enough. We can frequently improve the accuracy of the construction by choosing for the direction of each segment of the polygon that direction which does not belong to the starting but to the central point of the corresponding interval (Figs. 20 and 21).

We mention here in passing that graphical integration (that is, finding the graph of a primitive F{x) of a function f(x) which itself is given by a graph) can also be performed by means of a mechanical device, the so-called integraph. In this mechanism, a pointer is moved along the given curve and a pen automatically traces one of the curves y = F(x) for which F'(x) = f(x). The indeterminacy of the constant of integration is expressed by a certain arbitrariness in the initial position of the instrument. For integrating devices, cf. B. Williamson, Integral Calculus, pp. 214-217 (Longmans); Dictionary of Applied Physics, Vol. Ill, pp. 460-467 (Macmillan, 1923).

In Fig. 21, the construction described above is carried out for the function f(x)=x. By graphical integration, we obtain an approximation to the integral curve, which is the parabola y = ½x² - ½. In addition, Fig. 20 shows an approximation to the integral function of the function f(x) = 1/x. We shall study this integral later in greater detail - it will turn out to be the logarithmic function. Finally, the reader would he well advised to work out some other examples on his own, e.g., the graphical integration of the functions sin x and cos x.

Exercises 2.4:

1. Construct by graphical integration with the interval h = 1/10 the following integral curves:

In particular, evaluate

2.6. Further Remarks on the Connection between the Integral and the Derivative

Before we pursue systematically the relationships found in 2.4, we shall illustrate them from another point of view, which is closely related to the intuitive idea of density and other physical concepts.

2.6.1 Mass Distribution and Density; Total Quantity and Specific Quantity: We assume that any mass is distributed along a straight line, the x-axis, the distribution being continuous, but not necessarily uniform. For example, we may think of a vertical column of air standing on a surface of area 1; we take as x-axis a line pointing vertically upwards and as origin the point on the Earth's surface. The total mass between two abscissae x1 and x2 is then determined in the following manner by means of a so-called sum-function F(x). We measure the distance along the line from the initial point of the mass-distribution x = 0 and denote by F(x) the total mass between the abscissa 0 and the abscissa x. The increment of mass from the abscissa x1 to the abscissa x2 is then given simply by

thus a sign is assigned to the increment and this sign changes if x1 and x2 are interchanged.

The average mass per unit length in the interval x1 to x2 is

If we assume that the function F(x) is differentiable, then, as x2 ® x1, this value tends to the derivative F'(x1). This quantity is precisely what is usually called the specific mass or density of the distribution at the point x1; as a rule, of course, its value depends on the particular point chosen. There exists accordingly between the density f(x) and the sum-function F(x) the relation

The sum-function is a primitive function of the density, or, what amounts to the same thing, the mass is the integral of the density; conversely, the density is the derivative of the sum-function.

Exactly the same relation is very frequently encountered in physics. For example, if we denote by Q(t) the total amount of heat needed to raise the unit mass of a substance from the temperature t0 to the temperature t, then to raise the temperature from t1 to t2 requires the amount of heat

Between t1 to t2, the average amount of heat used per unit increase in temperature is then

If we assume once again differentiability of the function Q(t), we obtain in the limit the function

which we call the specific heat of the substance. In general, this specific heat is to be regarded as a function of the temperature. Here again, there exists between the specific heat and the total quantity of heat the characteristic relationship of integral and derivative

We shall encounter the same relations in all cases where total and specific quantities are interrelated, e.g., electric charge with density of charge, or total force acting on a surface with force-density or pressure.

In Nature, usually what we know directly is not density or specific quantity, but total quantity, whence it is the integral which is primary (as the name primitive suggests) and the specific quantity is only arrived at after a limiting process, namely, differentiation.

Incidentally, it may be noted that if the masses considered are by their nature positive, the sum-function F(x) must be a monotonically increasing function of x, and consequently the specific quantity, the density f(x), must be non-negative. However, nothing stops us from considering also negative quantities (for example, negative electricity); then our sum-functions F(x) need no longer be monotonic.

2.6.2 The Question of Applications: Perhaps, the relationship of the primitive sum-function to the density distribution becomes clearer when it is realized that, from the point of view of physical facts, the limiting processes of integration and differentiation represent idealizations and that they do not express anything exact in nature. On the contrary, in the realm of physical reality, we can form in place of the integral only a sum and in place of the derivative only a difference quotient of very small quantities. The quantities Dx remain different from 0; the passage to the limit Dx®0 is merely a mathematical simplification, in which the accuracy of the mathematical representation of the reality is not essentially impaired.

As an example, we return to the vertical column of air. According to the atomic theory, we find that we cannot think of the mass distribution as a continuous function of x. On the contrary, we will assume (and this, too, is a simplifying idealization) that the mass is distributed along the x-axis in the form of a large number of point-molecules lying very close to each other. Then the sum-function F(x) will not be continuous, but it will have a constant value in the interval between two molecules and will take a sudden jump as the variable x passes the point occupied by a molecule. The amount of this jump will be equal to the mass of the molecule, while the average distance between molecules, according to results established in atomic theory, is of the order of 10-8 cm. Now, if we are performing upon this air column some measurement in which masses of the order 104 molecules are to be considered negligible, our function cannot be distinguished from a continuous function. In fact, if we choose two values x and as x + Dx, the difference Dx of which is less than 10-4 cm, then the difference between F(x) and F(x+ Dx) will be the mass of the molecules in the interval; since the number of these molecules is of the order of 104, the values of F(x) and F(x+ Dx) are equal as for as our experiment is concerned. We consider simply as density of distribution the difference quotient

it is an important physical assumption that we do not obtain measurably different values for this quotient when Dx is allowed to vary between certain bounds, say between 10-4 and 10-5 cm. Now, imagine that F(x) is measured and plotted for a large number of points about 10-4 cm. apart and that the points thus found are joined by straight lines; we obtain a polygon and, by rounding off the corners, obtain finally a curve with a continually turning tangent. This curve is the graph of some function, say F1(x). This new function cannot be distinguished within the limits of experimental accuracy from F(x) and its derivative is within the same limits equal to DF/Dx; we thus have found a continuous differentiable function which for the purposes of physics is the function F(x).

It is perhaps appropriate to discuss yet another example of the concepts of sum-function and distribution density. In statistics, e.g., in the kinetic theory of matter or in statistical biology, these concepts frequently occur in a form in which the nature of the mathematical idealization is particularly clear. For example, let us consider the molecules of a gas confined in a vessel and observe their velocities at a given instant of time. Let the number of molecules be N and the number of those with velocities less than x be NF(x). Then F(x) denotes the ratio of the number of molecules moving with velocities between 0 and x to the total number of molecules. Of course, this sum-function is not continuous, but is sectionally constant (cf. Chapter IX) and suddenly increases by 1/N when x, as it increases, passes a value which is equal to the velocity of some molecule.

The idealization which we shall make here is that we shall think of the number N as increasing beyond all bounds. We assume that, in this passage to the limit N ® ¥, the sum-function F(x) tends to a definite continuous limit function F(x). That this is really the case, i.e., that we can with sufficient accuracy replace F(x) by this continuous function F(x), is obviously an important physical assumption; and it is another such assumption to assume that this sum-function F(x) possesses a derivative F '(x) = f(x), which we then call the density-distribution. The sum-function is connected with the density distribution by the equations

The density distribution is occasionally referred to as the specific probability that a molecule possesses the velocity x. The idealization we have just carried out has a great role in the kinetic theory of gases of Maxwell; it appears in exactly the same mathematical form in many problems of mathematical statistics.

2.7 The estimation of Integrals and the mean Value Theorem of the Integral Calculus

We close this chapter with some considerations of a matter of general significance, the full importance of which will not appear until somewhat later on. The point in question is the estimation of integrals.

2.7.1 The Mean Value Theorem of the Integral Calculus: The first and simplest of these estimation roles runs as follows: If in an interval a £ x £ b the continuous function f(x) is everywhere non-negative (is either positive or zero), then the definite integral

is also non-negative. Similarly, the integral is not positive, if the function is nowhere positive in the interval. The proof of this theorem follows directly from the definition of the integral. This leads to the theorem: If

everywhere in the interval a £ x £ b, then also

In fact, by our first remark, the integral of the difference f(x) - g(x) is non-negative and, by our addition rule,

Let M be the greatest and m the smallest value of the function f(x) in the interval ab. The function M-f(x) is non-negative in the interval and the same is true for the function f(x)-m, whence we obtain immediately the double inequality

However,

and likewise

whence

Hence the integral under consideration can be represented as the product of (b-a) and some number m between m and M:

As a rule, there is no need to state the exact value of this mean value m. However, we may say that it will be assumed by the function at least at one point x of the interval a £ x £ b, since in its interval of definition a continuous function assumes all values between its greatest and smallest values. As in the case of the mean value theorem of the differential calculus, the exact statement of the value x is in many cases unimportant. Hence we may set m = f(x), where x is an intermediate value of x, and find then

This last formula is called the mean value theorem of the integral calculus.

We can generalize the theorem somewhat by considering instead of the integrand f(x) an integrand of the form f(x)p(x), where p(x) is an arbitrary/ non-negative function which, like f(x), is assumed to be continuous. Since mp(x) £ f(x)p(x)£ Mp(x), we find immediately

or, as a single equation,

where x is again a number between a and b.

Thus, we have proved the theorem:

If f(x) and p(x) are continuous functions in a £ x £ b and p(x) ³ 0, then

where a £ x £ b.

2.7.2 Applications. The Integration of xa for any Irrational Value of a: The mean value theorem and the equivalent integral estimates give us immediately an insight into an intuitive and easily understood fact: The value of an integral changes very little, if the function itself is everywhere changed very little. In precise language: If in the entire interval a £ x £ b the absolute value of the difference of two functions f(x) and g(x) is less than e, then the absolute value of the difference of their integrals is less than e (b - a). In symbols: If |f(x - g(x) < e throughout the interval a £ x £ b, then

or

Fig. 22 illustrates very clearly this theorem. We draw for the curve y = f(x) the parallel curves y=f(x)+e and y=f(x)-e. By assumption, the function g(x) keeps within the strip bounded by these parallel curves. It is clear from this that the areas, which are bounded by the curves f(x) and g(x), differ from each other by less than half the area of the strip, and the area of the strip is just

There is no need for intuition. Since

it follows, by arguments analogous to those in 2.7.1,

which, as the result of the fundamental rules of integration, takes the form

here we have merely replaced the integral of a sum by the corresponding sum of integrals and have taken into consideration that

As an indication of the importance of this theorem, we shall show that with its help we can integrate the function xa for any irrational value of a, or more precisely, calculate the definite integral Here we assume that 0 < a < b.

We represent the index a as the limit of a sequence of rational numbers a1, a2, ··· ,an, ··· so that we can here assume that none of the values an is equal to -1, since an itself is different from -1. Now, we use for the power xa the definition

and note that, no matter how small a positive number e we choose, we can always find an n so large that in the entire interval t £ x £ b * we have

* This can be proved quite simply as follows (cf. A1.3). Remembering that xa is monotonic and setting dn = an - a, we have

in fact, xa lies between aa and ba, so that xa £ aa + ba, and likewise lies between Fromit follows that

hence, if n is chosen large enough, the right-hand side of the inequality is less than e. This yields simultaneously for all values of x in the interval a £ x £ b.

Now we need only apply the relationship, referred to above, to the functions f(x) = xa and , yielding

However, the integrals on the right-hand and left-hand sides may be evaluated in accordance with. earlier results, which yields

If we now let the number e decrease steadily and tend to 0, the corresponding values of n increase beyond all bounds; the number smust then converge to a, aa and ba, respectively, and we immediately obtain the result

In other words, the integration formula, which holds for rational values of a, also holds for irrational values of a.

It follows from this, by virtue of the fundamental theorem, that, for positive values of x, the differentiation formula for rational values

is also valid for irrational values of a.

Exercises 2.5:

1. Find the intermediate value x of the mean value theorem of the integral calculus for the following integrals and interpret them geometrically:

2. Let f(x) be continuous. Prove, using the mean value theorem of the integral calculus, that the derivative of the indefinite integral of f(x) is equal to f(x).

3. (a) Evaluate What is Interpret it geometrically. (b) Do the same for

4.* Let the function f(x) be continuous for all values of x and let F(x) be defined by

where d is an arbitrary positive number. Prove that:

(a) the function F(x) possesses a continuous derivative for all values of x,
b)
in any fixed interval a £ x £ b, we can make |F(x) - f(x)| < e, where e is an arbitrary pre-assigned positive number, by choosing d small enough.

5. Schwartz's Inequality for integrals: Prove that for all continuous functions f(x), g(x)

Appendix to Chapter II

A2.1 The Existence of the Definite Integral of a Continuous Function: We must still give a proof of the fact that there always exists the definite integral of a continuous function between the limits a and b (a < b). For this purpose, we recall the earlier discussed notation and consider the sum

It is certainly true that

where f(vn) denotes the least and g(un) the largest value of the function in the n-th sub-interval. The problem is to prove that Fn tends to a definite limit independently of the particular manner of subdivision and of the particular choice of the quantities xn , provided that, as n increases, the length of the longest subinterval tends to zero. In order to establish this, it is obviously necessary and sufficient to show that the two expressions converge to one and the same limit.

No matter how small the positive number e is chosen, we know from the uniform continuity of f(x) that in every sufficiently small interval the oscillation |f(un) - f(vn)| is less than e so that, if the subdivision is fine enough, we certainly must have

Hence we see that, as n increases, this difference must tend to zero, and so we can be content with proving that one of the sums, say , converges. This convergence will have been proved as soon as we show that can be made as small as desired by requiring that the corresponding subdivisions (which we shall refer to as subdivision n and subdivision m, respectively, exceed a certain degree of fineness. This degree of fineness is characterized by the property that for both subdivisions the oscillation of the function in each subinterval is less than e (e>0). We continue to a third subdivision the points of subdivision of which consist of all the points of subdivision n and of subdivision m taken together. This new subdivision, which has, say l points of subdivision, we denote by the subscript l and consider the corresponding upper sum . We shall now estimate the value of ,first obtaining estimates for the expressions . We assert that the following two relationships hold:

The proof follows at once from the meaning of our expressions. Let us consider, say, the n-th subinterval of the subdivision n. This subinterval will consist of one or several subintervals of the subdivision l; the terms corresponding to these intervals will each consist of two factors, one of which is a difference Dx and the other certainly not greater than f(un) and not less than f(vn). The sum of the lengths Dx of those intervals of the subdivision l which lie in the n-th subinterval of the coarser subdivision n is, however, exactly Dxn. Hence we see that the corresponding contribution to the sum must lie between the limits f(un)Dxn and f(vn)Dxn. If we now sum over all the n subintervals, we obtain the first of the above inequalities; the second is obtained in exactly the same manner, if we consider the subdivision m instead of the subdivision n.

We have already seen that it is likewise true that Hence,by the inequalities for proved above, one has

Thus, it is also certain that

Since we can choose e as small as we please, this relation shows, by Cauchy's convergence test, that the sequence of numbers actually converges. At the same time, we see at once from our argument that the limiting value is completely independent of the manner of sub-division.

The proof of the existence of the definite integral of a continuous function is thus complete.

Our method of proof teaches us yet more. It shows us that, in many cases, we are also led to the integral by a somewhat more general limiting process. If, for example, f(x) = f(x)y(x) and the interval from a to b is divided into n parts by the points xn , we consider instead of the sum Sf(xn)Dxn the more general sum

where x 'n and x "n are two not necessarily coincident points of the n-th subinterval. This sum will also tend to the integral

as n increases, provided that the length of the longest subinterval tends to zero. A corresponding statement holds for all sums formed in an analogous manner; for example, the sum

tends to the integral

The proof of these facts follows along exactly the same lines like those of the above proof and hence need not be worked out in detail.

A2.2 The Relation between the Mean Value Theorem of the Differential Calculus and the Mean Value Theorem of the Integral Calculus: There exists between the mean value theorems of the differential and the integral calculus a simple relation which is arrived at by way of the fundamental theorem and which we give as an instructive example of the use of that theorem. We take the mean value theorem of the integral calculus in its more special form

If we set put so that f(x) = F'(x), this formula assumes the form

Obviously, we can choose here for F(x) any function the first derivative F'(x)=f(x) of which is continuous, and thus, for such functions, the mean value theorem of the differential calculus has been proved.

If we consider the more general form of the mean value theorem of the integral calculus

where p(x) is a function which in our interval is continuous and positive and f(x) is an arbitrary continuous function, we are led to a correspondingly more general mean value theorem of the differential calculus. We set

and

the above mean value formula then takes the form

or, since f(x) = F'(x)/G'(x),

where a ¹ b.

This formula, in which x once again denotes a number between a and b, is the generalized mean value theorem of the differential calculus. For this to be valid, it is obviously sufficient to assume that F(x) and G(x) are continuous functions with continuous first derivatives and that, in addition, G'(x) is everywhere positive (or everywhere negative). In fact, with these assumptions, the entire process can be reversed.

Finally, it should be observed that in the present discussion of the mean value theorem of the differential calculus we have had to make assumptions more stringent than the theorems themselves require. (cf. 2.3.8 and 3.3.3)

Exercise 2.6:

1. Show that if f(x) has a continuous derivative in the interval a £ x £ b, then f(x) can be represented as the difference of two monotonic functions.