INTEGRAL CALCULUS.

CHAPTER I.

THE THEORY OF DEFINITE AND INDEFINITE INTEGRATION.

ARTICLE I.] The primary problem of the Integral Calculus is the summation of a series of which each term is an infinitesimal, and the number of terms is infinite; all the terms being infinitesimals of the same order, and the difference between two consecutive terms being an infinitesimal of an order higher than that of each term. Thus, according to the doctrine of infinitesimals and infinities, which has been established in Vol. I, if relatively to a given base the orders of infinity and of infinitesimal are the same, the sum will be finite; and as the order of infinity is higher or lower than that of the infinitesimal, so will the sum be infinite or infinitesimal. In most of the cases which will hereafter come under discussion, the sum will be finite; yet not in all and the quality and the circumstances of those infinitesimals, the sum of an infinite number of which is not finite, will require most cautious and careful consideration.

The infinitesimal term of the series will be expressed as a function of one or more variables; and the variation of the variables will give the several and successive terms of the series. Thus, f(x) dx, f(x, y)dxdy,.............. may be general terms of such a series; and the successive terms will be given by means of the continuous variations of the variables. These are the most general forms of the infinitesimal terms; x" dx, cosx dx, eax+by dx dy are particular forms. Such infinitesimal terms are called infini PRICE, VOL. II.

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tesimal elements, being the elements or infinitesimal parts of the whole sum; and the finite factor in each is called the elementfunction. The form of these latter factors evidently assigns the law of the series. In this respect these series are analogous to those ordinary series each of whose terms is a finite quantity, and in which the general term is a type-term, and assigns the law of the series.

The problem of the summation of a series is generally indefinite; but becomes definite when two terms are assigned, the sum of the terms within which, either inclusively or exclusively, is to be found. These assigned terms are called the limiting terms, or the limits of the series; the first and last terms being called respectively the inferior and the superior limit. The excess of the superior over the inferior limit is called the range of summation. Although the sums of some series can be found for any general limits, say the mth and the nth terms, yet the sums of others can be found only for certain specified limits. These peculiarities depend on the law of the series, and many instances of them will occur hereafter.

Although the number of terms in the series, the sum of which is required to be found in the problem of the Integral Calculus, is infinite; yet as they are infinitesimal, and vary by infinitesimal variations of the subject variables which express their general terms, the difference between the values of the variables at the limits will be a finite quantity; so that the problem is definite as to its limits, and the distance between the limits is finite.

The process by which such sums are found is termed Integration, being, as it is, the putting together the parts of which a whole is composed; and the sum of the series of infinitesimal elements between given terms is called a Definite Integral, the values of the variable which assign the first and last terms being called the limits of Integration; and the excess of the superior limit over the inferior is called the range of integration. As the form of each term of the series is the same, if a general term is given, the general form of the sum of a series of terms can in many cases be found, although the first and last terms may not be given; this general sum is called the Indefinite Integral.

The Integral Calculus is the aggregate of the rules by which Integrals are determined, and the code of laws subject to which Differentials and Integrals in their mutual relations may be applied to questions of Geometry and Physics.

In the early part of the treatise I shall consider functions of one variable; and I shall assume the infinitesimal element to be of the form f(x)dx, and the superior and inferior limits to be respectively, and xo; so that x-xo is the range of integration. I shall also assume f(x) to be finite and continuous for all values of a between a, and x: we shall hereby avoid difficulties as to discontinuity and infinite values of functions. Thus the simple problem of the Integral Calculus is to find the sum of an infinite number of infinitesimal elements as a increases by infinitesimal increments from a to r; and it takes the following form;

Let - be divided into n infinitesimal parts, and let x,, x2,.. x-1 be the values of a corresponding to the points of partition. Let s be the definite integral; then observing that f(x) is multiplied by the increment immediately succeeding a, we have S = f(x) (x1−x ̧) +ƒ (X1) (X2−X1) + ... + ƒ (xn−1) (X„ −xn−1); (1) and xn

f

and we have to find the value of S in terms of

xo

2.] Let us look at the theory from another point of view, and consider the genesis or origin of element-functions as it is presented to us in the Differential Calculus.

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x2;

2

Let us take the following problem; let a be the length of a line OP (see Fig. 1) which varies continuously from OP。。 up to OP, ; on OP, OP, OP, let squares be described, viz. OR, OR, OR, So that OR = let op be increased by an infinitesimal PQ dx, and on oq let a square be described; then the increase of 2 due to the infinitesimal increase of x is 2x dx: suppose a similar process of augmentation to be performed on all values of x from x up to x; the effect of this will be that the square x2 will grow into the square a,2 by infinitesimal augments, each of which is of the form 2x dx, wherein a receives the successivelyincreased values. From another point of view however the effect of such a process is, to resolve the finite gnomonic area OR-OR into infinitesimal elements, which are infinitesimal gnomons, each being of the form PR's, which is expressed by 2x da; thus x2 will be resolved into elements 2x dx, corresponding to values of x from x = x up to x = x; and if x O, the whole square OR, will be resolved into its gnomonic infinitesimal elements. Or consider the following more general problem: Let F(x) be a function of a finite and continuous for all values of a between a and r; and let the difference x-xo be divided into n equal and

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finite parts each of which is equal to ax, so that x„−x = nâx; then by equation (21), Art. 116, of Vol. I,

F(x+x)-F(x) = ▲x F′(x+0sx)

F(x+2▲x) − F(x ̧+▲x) = ▲x F' (x + ▲ x +0 ▲x)

F(x+n^x) − F{x+(n−1)▲x} = ▲xF′{x+(n−1)ax+0ax}.. Let ar become infinitesimal, that is, become da; then adding the members of (2), and bearing in mind that x2 = x+ndx,

F(x) F(x) F′(x) dx + F'(x+dx) dx+F′(x + 2 dx) dx+... — = F′(x ̧)dx+F'′(x。+dx) ...+F′(xn−dx)dx; (3)

that is, the process of growth by infinitesimal increase, on which principle equations (2) are constructed, is equivalent to the resolution of F(x) — F(x) into infinitesimal elements, as exhibited in the right-hand member of equation (3).

(2)

Thus the Differential Calculus is a method by which a given finite function is resolved into its infinitesimal component elements; these being of such a nature, that the aggregate of an infinity of them is required to constitute the finite quantity. Or if the original function is an infinitesimal function, the elements into which it is resolved are infinitesimals of a higher order. The general form of all the elements is the same, as appears from the above examples, and therefore any one is a type of all, and expresses all but the form of the typical element varies, as the function varies, the law of connexion depending on the process of Differentiation. The subject then on which Differentiation is performed is the function, and the result is the resolution of that function into its elements. The process of Differentiation is therefore one of Disintegration.

3.] In the Integral Calculus the data are changed; an infinitesimal element is given, which is the type of all; and the sum of these between certain given limits is to be determined. The process then is the reverse of differentiation, and is that of summation, as before observed: I propose to illustrate it at first by two or three simple examples.

Let us suppose the element to be 2x dx; and the sum of all such to be required, as a continuously increases from x。 to xn; for the sake of simplicity, let x-xo be divided into n equal infinitesimal parts, each of which = i; so that

so that the definite integral of 2x dx is x‚2 —x2.

i}

The problem given in the preceding Article exhibits the meaning of the process from a geometrical point of view. 2x dx expresses the infinitesimal gnomon contained between the squares described on OP and on oq, and is consequently the infinitesimal element of 2 and the sum of all such gnomons between the limits and x is evidently x2-x2. If the inferior limit = 0, then the sum = x2. This latter result is also true, whatever is the value of x; consequently if x, has the general value x, x2 is the indefinite integral of 2x dr. As this mode of interpretation is important in giving body to our thoughts, let us take another example; in fig. 16, E is a point (x, y) in the plane of xy referred to the rectangular axes, ox and oy. H is a point (x+dx, y+dy) infinitesimally near to E, so that EG = dx, EF = GH =dy; consequently the infinitesimal rectangular area EH = dxdy. Now this rectangle is an infinitesimal element of the plane superficies; and is an element of a plane area bounded by any lines in the plane of xy. Thus the whole area will be the integral of these infinitesimal area-elements. If no limits of the area are given, the problem is evidently indefinite. Suppose however the area to be limited; and that it is required to express, say, the area м ̧3 ̧„„ in terms of an and C0 2 where омо = X。, OM2 = xn, and the equation to the bounding curve is y = f(x). We must find the sum of all the elements, similar to EH, contained between the two ordinates MP and NQ. Now as MP = f(x), NQ = f (x + dx), dx MN = dx; consequently MPQN = 2 = f(x) dx, since da is infinitesimal; and the whole area is the sum of all the similar elemental slices which are contained between the two ordinates Mo Po and M, P. And to find the sum of f(x)dx between these given limits is a problem of the Integral Calculus.

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{f(x)+f(x+dx)}, which