**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

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 D*y* = *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., *x*_{1} = -1, *x*_{2}*
=* 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 *x*_{0} be a
point in the interval between *x* and *x* + *h,*
at which the function *f*(*x*) takes its greatest
value, and *x*_{1} 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*(*x*_{0}) and
*hf*(*x*_{1}) which represent the areas of
rectangles with the interval from *x *to *x+h* as base
and the altitudes *f*(*x*_{0}) and *f*(*x*_{1}),
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 *u*_{0} = *x*,
*u*_{1}, *u*_{2}, ··· , *u*_{n}*
= 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 D*u*_{n}=*u*_{n} - *u*_{n}_{-1} tends to zero as *n* increases. Then D*u*_{n}/*h
*is certainly positive, no matter whether *h* is
positive or negative. Since we know that

and the sum of the quantities D*u*_{n}
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*(*x*_{0}) and *f*(*x*_{1})
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 F_{1}(*x*) and *F*_{2}(*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 *F*_{1}(*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 *F*_{1}(*x*)
and *F*_{2}(*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*(*x*_{1}) = *G*(*x*_{2})*.*
Since *x*_{1} and *x*_{2} 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 2*x*^{3/2}/3
by addition of a constant which is less than or equal to zero,
namely, by the constant -2*a*^{3/2}; yet such a
function as 2*x*^{3/2} + 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 D*x*_{1} D*x*_{2}, ··· , D*x*_{n}
and consider the sum S(D*F*/D*x*_{n})D*x*_{n}*.* On the one hand, this sum is simply SD*F* = *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/D*x*_{n} = *F* '(*x*_{n}), where *x*_{n} is a point between the ends *x*_{n-1} and *x*_{n}, of the interval D*x*. The sum is therefore equal to SD*x*_{n}*F*
'(*x*_{n}) 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.

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 *dx*^{a+1}/*dx* = (*a**+*1)*x*^{a}, whence
*x*^{a+1}/(*a**+*1) is a primitive function or indefinite
integral of *x*^{a}, provided that *a
*¹ -1, and thus, by 2.4.4 arrive at the above
integration formula.

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, *x*_{1},* x*_{2},
··· 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* *=* *x*_{1},
we draw the line with the slope *f*(*x*_{1});
through the intersection of this line with *x* *=* *x*_{2},
we draw the line with the slope *f*(*x*_{2}),
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* = ½*x*² - 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*.

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 *x*_{1} and *x*_{2}
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 *x*_{1} to
the abscissa *x*_{2} is then given simply by

thus a sign is assigned to the increment and
this sign changes if *x*_{1} and *x*_{2}
are interchanged.

The average mass per unit length in the
interval *x*_{1} to *x*_{2} is

If we assume that the function *F*(*x*)
is differentiable, then, as *x*_{2 }® *x*_{1},
this value tends to the derivative *F*'(*x*_{1}). This
quantity is precisely what is usually called the **specific mass** or **density** of the distribution at the point *x*_{1};
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 *t*_{0} to the
temperature *t,* then to raise the temperature from *t*_{1}
to *t*_{2} requires the amount of heat

Between *t*_{1} to *t*_{2},
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 D*x* remain
different from 0; the passage to the limit D*x*®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 10^{4} 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* + D*x*, the difference D*x* of which
is less than 10^{-4} cm, then the difference between *F*(*x*)
and *F*(*x+* D*x*) will be the mass of the molecules in the
interval; since the number of these molecules is of the order of
10^{4}, the values of *F*(*x*) and *F*(*x+*
D*x*)
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 D*x* 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 *F*_{1}(*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 D*F*/D*x*; 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 *N**F*(*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 ***x*^{a}** 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 *x*^{a} 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 *a*_{1}, a_{2}*, *··· ,*a*_{n}*,
*··· so that we can here assume that
none of the values *a*_{n} is equal to -1, since *a*_{n}
itself is different from -1. Now, we use for the power *x*^{a}
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 *x*^{a}
is monotonic and setting *d*_{n }= a_{n} - a, we have

in fact, *x*^{a}
lies between *a*^{a} and *b*^{a}, so that *x*^{a}^{ }£ *a*^{a} + *b*^{a}, 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*) = *x*^{a} 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*, *a*^{a}
and *b*^{a},
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*.

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

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

**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*(*v*_{n})
denotes the least and *g*(*u*_{n}) the
largest value of the function in the *n*-th sub-interval.
The problem is to prove that *F*_{n}
tends to a definite limit independently of the particular manner
of subdivision and of the particular choice of the quantities *x*_{n} , 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*(*u*_{n})
- *f*(*v*_{n})| 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 D*x* and the other
certainly not greater than *f*(*u*_{n})
and not less than *f*(*v*_{n}). The
sum of the lengths D*x* of those intervals of the subdivision *l*
which lie in the *n*-th subinterval of the coarser subdivision *n* is,
however, exactly D*x*_{n}. Hence we see that the corresponding contribution to
the sum must lie between the limits *f*(*u*_{n})D*x*_{n}
and *f*(*v*_{n})D*x*_{n}. 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 *x*_{n }, we consider instead of the sum S*f*(*x*_{n})D*x*_{n} 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)

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.