Integral calculus

“ From the considerations indicated in this chapter, we see that, while the differential calculus constitutes by its nature a limited and perfect system, to which nothing essential remains to be added, the integral calculus, or the simple system of integration, presents necessarily an inexhaustible field for the activity of the human mind, independently of the indefinite applications of which the transcendental analysis is evidently susceptible. ”

“ Any process for evaluating a definite integral is a process of integration, and the rules for evaluating integrals constitute the integral calculus.
6. The chief of these rules is obtained by regarding the extreme ordinate BD as variable. Let ξ now denote the abscissa of B. The area A of the figure ACDB is represented by the integral , and it is a function of ξ. Let BD be displaced to B′D′ so that becomes (seeTheorem of Inversion. fig. 4) . The area of the figure ACD′B′ is represented by the integral and the increment ΔA is given by the formula:
which represents the area BDD′B′. ”

“ Finally, to complete our review of the vast collection of analytical researches of which is composed the integral calculus, properly so called, there remains to be mentioned one theory, very important in all the applications of the transcendental analysis, which I have had to leave outside of the system, as not being really destined for veritable integration, and proposing, on the contrary, to supply the place of the knowledge of truly analytical integrals, which are most generally unknown. ”

“ Thus, for example, in the case of rectifications, it will be first necessary to calculate dy, or dy and dz, by differentiating the equation or the equations of each curve proposed; after eliminating these expressions, the general differential formula above enunciated will then contain only ds and dx; having arrived at this point, the elimination of the infinitesimals can be completed only by the integral calculus. ”

“ Besides the method of finding the integrals of given differentials, the integral calculus is also applied to various branches of mathematics, as well as to almost every branch of natural philosophy and engineering. ”

“ They are those in which the differential equations are found to be immediately ready for integration, because they contain, at their first formation, only the infinitesimals which relate to the functions sought, or to the really independent variables, without its being necessary to introduce, differentially, other functions as intermediaries. If in these new cases we introduce these last functions, since, by hypothesis, they will enter directly and not by their differentials, ordinary algebra will suffice to eliminate them, and to bring the question to depend on only the integral calculus. ”

“ And when such a differential equation has been found, one can get no further until it has been integrated. Generally it is much easier to state the appropriate differential equation than to solve it:–the real trouble begins then only when one wants to integrate, unless indeed the equation is seen to possess some standard form of which the integral is known, and then the triumph is easy. ”

“ Any simple integral can be regarded as a line-integral taken along a portion of the axis of x. When a change of variables is made, the limits of integration with respect to the new variable must be such that the domain of integration is the same as before. This condition may Line Integrals and Multiple Integrals. require the replacing of the original integral by the sum of two or more simple integrals.
(ii.) The line integral of a perfect differential of a one-valued function, taken along any closed curve, is zero. ”

“ The integration of a rational function is generally effected by resolving the function into partial fractions, the function being first expressed as the quotient of two rational integral functions. Corresponding to any simple root of the denominator there is a logarithmic term in the integral. If any of the roots of the denominator are repeated there are rational algebraic terms in the integral. ”

“ If then we were to subtract the smaller area from the larger, we should have left as a remainder the area , which is what we want. Here we have the clue as to what to do; the definite integral between the two limits is the difference between the integral worked out for the superior limit and the integral worked out for the lower limit.
Let us then go ahead. First, find the general integral thus:
.
and, as is the equation to the curve (Fig. 52) ,
is the general integral which we must find. ”

“ The formation of the first variation of an integral by means of a weak variation can be carried out without difficulty in the case of a simple integral involving any number of dependent variables and differential coefficients of arbitrarily high orders, and also in the cases of double and multiple integrals ”

“ When an integral is to be thus defined between two limits, we call the lower of the two values the inferior limit, and the upper value the superior limit. Any integral so limited we designate as a definite integral, by way of distinguishing it from a general integral to which no limits are assigned. ”

“ Abel's theorem was pronounced by Jacobi the greatest discovery of our century on the integral calculus. The aged Legendre, who greatly admired Abel's genius, called it "monumentum aere perennius. During the few years of work allotted to the young Norwegian, he penetrated new fields of research, the development of which has kept mathematicians busy for over half a century. ”

“ Induction is, in fact, the inverse operation of deduction, and cannot be conceived to exist without the corresponding operation, so that the question of relative importance cannot arise. Who thinks of asking whether addition or subtraction is the more important process in arithmetic? But at the same time much difference in difficulty may exist between a direct and inverse operation; the integral calculus, for instance, is infinitely more difficult than the differential calculus of which it is the inverse. ”

“ We have two theorems of transformation connecting volume integrals with surface integrals and surface integrals with line integrals. The first theorem, called “Green’s theorem,” is expressed by the equationTheorems of Green and Stokes.
where the volume integral on the left is taken through the volume within a closed surface S, and the surface integral on the right is taken over S, and l, m, n denote the direction cosines of the normal to S drawn outwards. There is a corresponding theorem for a closed curve in two dimensions, viz.,
the sense of description of s being the positive sense. ”

“ The mathematical questions which have given birth to the Calculus of Variations consist generally in the investigation of the maxima and minima of certain indeterminate integral formulas, which express the analytical law of such or such a phenomenon of geometry or mechanics, considered independently of any particular subject. ”

“ XIII., p. 122, here, can be made use of in integration.
Take again ; if we split into partial fractions, this becomes (see p. 232) :
.
Notice that the same integral can be expressed sometimes in more than one way (which are equivalent to one another) .
Pitfalls. A beginner is liable to overlook certain points that a practised hand would avoid; such as the use of factors that are equivalent to either zero or infinity, and the occurrence of indeterminate quantities such as . There is no golden rule that will meet every possible case. Nothing but practice and intelligent care will avail. ”

“ It is a fundamental theorem that this equation cannot hold for all admissible functions u, unless the differential equation Li.. 'f§ f. Q§ =0
dx éy' 6y
is satisfied at every point of the curve along which the integral is taken. This is the principal equation for this problem. The curves that are determined by it are called the stationary curves, or the extremals, of the integral. We learn that curves the integral cannot be an extremum unless it is taken along a stationary curve. ”

“ No one, even to-day, is able to find the general integral of the expression,
,
because has never yet been found to result from differentiating anything else.
Another simple case.
Find .
On looking at the function to be integrated, you remark that it is the product of two different functions of . You could, you think, integrate by itself, or by itself. Of course you could. But what to do with a product? None of the differentiations you have learned have yielded you for the differential coefficient a product like this. ”

“ The meaning of integration of a function of n variables through a domain of the same number of dimensions is explained in the article Function. In the case of two variables x, y we integrate a function ƒ (x, y) over an area; in the case of three variables x, y, z we integrate a function ƒ (x, y, z) Multiple Integrals. through a volume. The integral of a function ƒ (x, y) over an area in the plane of (x, y) is denoted by
The notation refers to a method of evaluating the integral. ”

“ There exist interesting investigations as to the connexion of the value of s above, the law of increase of the modulus of the integral function ƒ (z) , and the law of increase of the coefficients in the series ƒ (z) = Σanzn as n increases (see the bibliography below under Integral Functions) . It can be shown, moreover, that an integral function actually assumes every finite complex value, save, in exceptional cases, one value at most. For instance, the function exp (z) assumes every finite value except zero (see below under § 21, Modular Functions) . ”

“ In fact, even if we could integrate every equation of the first order relating to a single function, that would not be sufficient for obtaining the final integral of an equation of any order whatever, inasmuch as every differential equation is not reducible to that of an immediately inferior order. ”

“ The symbolical method which has been explained has wider applications than that to which we have, for simplicity of explanation, restricted it. For example, if ψ (x) be any function of x, and a1, a2, ... an be different constants, and [ (t + a1) (t + a2) ... (t + an) ] −1 when expressed in partial fractions be written Σcm (t + am) −1, a particular integral of the differential equation (D + a1) (D + a2) ... ”

“ When λ1 is a positive integer, not zero, the second solution about x = 0 persists as a solution, in accordance with the order of arrangement of the roots of the index equation in our theory; the first solution is then replaced by an integral polynomial of degree -λ or −μ1, when λ or μ is one of the negative integers 0, −1, −2, ..., 1 − λ1, but otherwise contains a logarithm. ”

“ If we except those cases in which the problem has no definite meaning because the data are such that the integral must be infinite, it is evident that a minimum must always exist, and (on account of the quadratic form of the integral) that it is unique. ”

“ In the language of the mathematician, the integrator gives only that miserly result, a definite integral, but the integraph yields an indefinite integral, a picture of the result at all times or all points—a much greater boon in most mechanical and physical investigations. ”

“ It is even certain that the last special improvements in mathematical analysis have contributed to extend our resources in that respect, by introducing within the domain of the calculus certain definite integrals, which in some respects supply the place of new simple functions, although they are far from fulfilling all the necessary conditions, which has prevented me from inserting them in the table of true analytical elements. ”

“ It is true that in the greatest number of important cases its use requires to be preceded and prepared by that of the Calculus of indirect Functions, which is intended to facilitate the establishment of equations. But, although algebra has then only a secondary office to perform, it has none the less a necessary part in the complete solution of the question, so that the Calculus of direct Functions must continue to be, by its nature, the fundamental base of all mathematical analysis. ”

“ To integrate a system of simultaneous equations of the first order X1 = a1, ... Xr = ar in n independent variables x1, ... xn and one dependent variable z, we write p1 for dz/dx1, &c., System of equations of the first order. and attempt to find n + 1 − r further functions Z, Xr+1 ... Xn, such that the equations Z = a, Xi = ai, (i = 1, ... n) involve dz − p1dx1 − ... − pndxn = 0. By an argument already given, the common integral, if existent, must be satisfied by the equations of the characteristic chains of any one equation Xi = ai ”