1911 Encyclopædia Britannica/Infinitesimal Calculus/Outlines 3
III. Outlines of the Infinitesimal Calculus (§40-46) | Infinitesimal Calculus III. Outlines of the Infinitesimal Calculus (§47-56) |
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III. Outlines of the Infinitesimal Calculus (§47-56)[edit]
47. The formal definition of an integral, the theorem of the existence of the integral for certain classes of functions, a list of classes of “integrable” functions, extensions of the notion of integration to functions which become infinite or indeterminate, and to cases in which the limits of integration Integral calculus. become infinite, the definitions of multiple integrals, and the possibility of defining functions by means of definite integrals—all these matters have been considered in Function. The definition of integration has been explained in § 5 above, and the results of some of the simplest integrations have been given in § 12. A few theorems relating to integrations have been noted in §§ 34, 35, 36 above.
48. The chief methods for the evaluation of indefinite
integrals are the method of integration by parts, and the
introduction of new variables.
Methods of integration.
From the equation d(uv) = udv + vdu we deduce the equation
or, as it may be written
This is the rule of “integration by parts.”
As an example we have
When we introduce a new variable z in place of x, by means of an equation giving x in terms of z, we express ƒ(x) in terms of z. Let φ(z) denote the function of z into which ƒ(x) is transformed. Then from the equation
we deduce the equation
As an example, in the integral
put x=sin z; the integral becomes
49. The indefinite integrals of certain classes of functions can be expressed by means of a finite number of operations of addition or multiplication in terms of the so-called “elementary” functions. The elementary functions are rational algebraic functions, implicit algebraic functions, exponentials Integration in terms of elementary functions. and logarithms, trigonometrical and inverse circular functions. The following are among the classes of functions whose integrals involve the elementary functions only: (i.) all rational functions; (ii.) all irrational functions of the form ƒ(x, y), where ƒ denotes a rational algebraic function of x and y, and y is connected with x by an algebraic equation of the second degree; (iii.) all rational functions of sin x and cos x; (iv.) all rational functions of e^{x}; (v.) all rational integral functions of the variables x, e^{ax}, e^{bx}, . . . sin mx, cos mx, sin nx, cos nx, . . . in which a, b, . . . and m, n, . . . are any constants. 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. The operation of resolving a fraction into partial fractions requires a knowledge of the roots of the denominator, but the algebraic part of the integral can always be found without obtaining all the roots of the denominator. Reference may be made to C. Hermite, Cours d’analyse, Paris, 1873. The integration of other functions, which can be integrated in terms of the elementary functions, can usually be effected by transforming the functions into rational functions, possibly after preliminary integrations by parts. In the case of rational functions of x and a radical of the form (ax^{2}+bx+c) the radical can be reduced by a linear substitution to one of the forms √(a^{2}−x^{2}), √(x^{2}−a^{2}), √(x^{2}+a^{2}). The substitutions x = a sin θ, x = a sec θ, x = a tan θ are then effective in the three cases. By these substitutions the subject of integration becomes a rational function of sin θ and cos θ, and it can be reduced to a rational function of t by the substitution tan 12θ = t. There are many other substitutions by which such integrals can be determined. Sometimes we may have information as to the functional character of the integral without being able to determine it. For example, when the subject of integration is of the form (ax^{4} + bx^{3}+cx^{2}+dx+e)^{−1/2} the integral cannot be expressed explicitly in terms of elementary functions. Such integrals lead to new functions (see Function).
Methods of reduction and substitution for the evaluation of indefinite integrals occupy a considerable space in text-books of the integral calculus. In regard to the functional character of the integral reference may be made to G. H. Hardy’s tract, The Integration of Functions of a Single Variable (Cambridge, 1905), and to the memoirs there quoted. A few results are added here
(i.)
(ii.) can be evaluated by the substitution x-p=1/z, and can be deduced by differentiating (n−1) times with represt to p.
(iii.) can be reduced by the substitution to the form
where A and B are constants, and λ_{1} and λ_{2} are the two values of λ for which (a−λα)x^{2}+2(b−λβ)x+c−λγ is a perfect square (see A. G. Greenhill, A Chapter in the Integral Calculus, London, 1888).
(iv.) ∫x^{m} (ax^{n}+b)^{p} dx, in which m, n, p are rational, can be reduced, by putting ax^{n} = bt, to depend upon ∫t^{q} (1+t)^{p}dt. If p is an integer and q a fraction r/s, we put t = u^{s}. If q is an integer and p = r/s we put 1+t = u^{s}. If p+q is an integer and p = r/s we put 1+t = tu^{s}. These integrals, called “binomial integrals,” were investigated by Newton (De quadratura curvarum).
(v.) (vi.)
(vii.) ∫
(viii.) ∫ can be reduced by differentiating a function of the form sin^{p} x cos^{p} x;
e.g.
Hence
(xi.) (n an integer)
(x.) (n an integer)
(xi.) can be reduced by one of the substitutions
of which the first or the second is to be employed according as e < or > 1.
50. Among the integrals of transcendental functions
which lead to new transcendental functions we may notice
New transcendents.
or
called the “logarithmic integral,” and denoted by “Li x,” also the integrals
and
called the “sine integral” and the “cosine integral,” and denoted by “Si x” and “Ci x,” also the integral
called the “error-function integral,” and denoted by “Erf x.” All these functions have been tabulated (see Tables, Mathematical).
51. New functions can be introduced also by means of the definite integrals of functions of two or more variables with respect to one of the variables, the limits of integration being fixed. Prominent among such functions are the Beta and Gamma functions expressed by the equationsEulerian integrals.
When n is a positive integer Γ(n + 1) = n!. The Beta function (or “Eulerian integral of the first kind”) is expressible in terms of Gamma functions (or “Eulerian integrals of the second kind”) by the formula
The Gamma function satisfies the difference equation
and also the equation
with the particular result
The number
or
is called “Euler’s constant,” and is equal to the limit
its value to 15 decimal places is 0.577 215 664 901 532.
The function log Γ(1 + x) can be expanded in the series
where
and the series for log Γ(1 + x) converges when x lies between −1 and 1.
52. Definite integrals can sometimes be evaluated when the limits
of integration are some particular numbers, although
the corresponding indefinite integrals cannot be found.
For example, we have the result
Definite integrals.
although the indefinite integral of (1−x^{2})^{−1/2} log x cannot be found. Numbers of definite integrals are expressible in terms of the transcendental functions mentioned in § 50 or in terms of Gamma functions. For the calculation of definite integrals we have the following methods:—
(i.) | Differentiation with respect to a parameter. |
(ii.) | Integration with respect to a parameter. |
(iii.) | Expansion in infinite series and integration term by term. |
(iv.) | Contour integration. |
The first three methods involve an interchange of the order of two limiting operations, and they are valid only when the functions satisfy certain conditions of continuity, or, in case the limits of integration are infinite, when the functions tend to zero at infinite distances in a sufficiently high order (see Function). The method of contour integration involves the introduction of complex variables (see Function: § Complex Variables).
A few results are added
(i.) | (1>a>0), |
(ii.) | (0<a or b<1), |
(iii.) | (a>1), |
(iv.) | |
(v.) | |
(vi.) | |
(vii.) | or according as α < or > 1, |
(viii.) | |
(ix.) | |
(x.) | |
(xi.) | |
(xii.) | |
(xiii.) | |
(xiv.) |
53. 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. We may suppose the area divided into a very large number of very small rectangles by lines parallel to the axes. Then we multiply the value of ƒ at any point within a rectangle by the measure of the area of the rectangle, sum for all the rectangles, and pass to a limit by increasing the number of rectangles indefinitely and diminishing all their sides indefinitely. The process is usually effected by summing first for all the rectangles which lie in a strip between two lines parallel to one axis, say the axis of y, and afterwards for all the strips. This process is equivalent to integrating ƒ(x, y) with respect to y, keeping x constant, and taking certain functions of x as the limits of integration for y, and then integrating the result with respect to x between constant limits. The integral obtained in this way may be written in such a form as
and is called a “repeated integral.” The identification of a surface integral, such as ∫∫ ƒ(x, y)dxdy, with a repeated integral cannot always be made, but implies that the function satisfies certain conditions of continuity. In the same way volume integrals are usually evaluated by regarding them as repeated integrals, and a volume integral is written in the form
Integrals such as surface and volume integrals are usually called “multiple integrals.” Thus we have “double” integrals, “triple” integrals, and so on. In contradistinction to multiple integrals the ordinary integral of a function of one variable with respect to that variable is called a “simple integral.”
A more general type of surface integral may be defined by taking an arbitrary surface, with or without an edge. We suppose in the first place that the surface is closed, or has no edge. We may mark a large number of points on the surface, and draw the tangent planes at all these points. These Surface Integrals. tangent planes form a polyhedron having a large number of faces, one to each marked point; and we may choose the marked points so that all the linear dimensions of any face are less than some arbitrarily chosen length. We may devise a rule for increasing the number of marked points indefinitely and decreasing the lengths of all the edges of the polyhedra indefinitely. If the sum of the areas of the faces tends to a limit, this limit is the area of the surface. If we multiply the value of a function ƒ at a point of the surface by the measure of the area of the corresponding face of the polyhedron, sum for all the faces, and pass to a limit as before, the result is a surface integral, and is written
The extension to the case of an open surface bounded by an edge presents no difficulty. A line integral taken along a curve is defined in a similar way, and is writtenLine Integrals.
where ds is the element of arc of the curve (§ 33). The direction cosines of the tangent of a curve are dx/ds, dy/ds, dz/ds, and line integrals usually present themselves in the form
or
In like manner surface integrals usually present themselves in the form
where l, m, n are the direction cosines of the normal to the surface drawn in a specified sense.
The area of a bounded portion of the plane of (x, y) may be expressed either as
or as
the former integral being a line integral taken round the boundary of the portion, and the latter a surface integral taken over the area within this boundary. In forming the line integral the boundary is supposed to be described in the positive sense, so that the included area is on the left hand.
53a. 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. This theorem is a particular case of a more general theorem called “Stokes’s theorem.” Let s denote the edge of an open surface S, and let S be covered with a network of curves so that the meshes of the network are nearly plane, then we can choose a sense of description of the edge of any mesh, and a corresponding sense for the normal to S at any point within the mesh, so that these senses are related like the directions of rotation and translation in a right-handed screw. This convention fixes the sense of the normal (l, m, n) at any point on S when the sense of description of s is chosen. If the axes of x, y, z are a right-handed system, we have Stokes’s theorem in the form
where the integral on the left is taken round the curve s in the chosen sense. When the axes are left-handed, we may either reverse the sense of l, m, n and maintain the formula, or retain the sense of l, m, n and change the sign of the right-hand member of the equation. For the validity of the theorems of Green and Stokes it is in general necessary that the functions involved should satisfy certain conditions of continuity. For example, in Green’s theorem the differential coefficients ∂ξ/∂x, ∂η/∂y, ∂ζ/∂z must be continuous within S. Further, there are restrictions upon the nature of the curves or surfaces involved. For example, Green’s theorem, as here stated, applies only to simply-connected regions of space. The correction for multiply-connected regions is important in several physical theories.
54. The process of changing the variables in a multiple integral, such as a surface or volume integral, is divisible into two stages. It is necessary in the first place to determine the differential element expressed by the product of the differentials of the first set of variables in terms of the differentials of the Change of Variables in a Multiple Integral. second set of variables. It is necessary in the second place to determine the limits of integration which must be employed when the integral in terms of the new variables is evaluated as a repeated integral. The first part of the problem is solved at once by the introduction of the Jacobian. If the variables of one set are denoted by x_{1}, x_{2}, . . ., x_{n}, and those of the other set by u_{1}, u_{2}, . . ., u_{n}, we have the relation
In regard to the second stage of the process the limits of integration must be determined by the rule that the integration with respect to the second set of variables is to be taken through the same domain as the integration with respect to the first set.
For example, when we have to integrate a function ƒ(x, y) over the area within a circle given by x^{2}+y^{2} = a^{2}, and we introduce polar coordinates so that x = r cos θ, y = r sin θ, we find that r is the value of the Jacobian, and that all points within or on the circle are given by a ≥ r ≥ 0, 2π ≥ θ ≥ 0, and we have
If we have to integrate over the area of a rectangle a≥x≥0, b≥y≥0, and we transform to polar coordinates, the integral becomes the sum of two integrals, as follows:—
55. A few additional results in relation to line integrals and multiple integrals are set down here.
(i.) 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.
(iii.) The area within any plane closed curve can be expressed by either of the formulae
or
where r, θ are polar coordinates, and p is the perpendicular drawn from a fixed point to the tangent. The integrals are to be understood as line integrals taken along the curve. When the same integrals are taken between limits which correspond to two points of the curve, in the sense of line integrals along the arc between the points, they represent the area bounded by the arc and the terminal radii vectores.
(iv.) The volume enclosed by a surface which is generated by the revolution of a curve about the axis of x is expressed by the formula
and the area of the surface is expressed by the formula
where ds is the differential element of arc of the curve. When the former integral is taken between assigned limits it represents the volume contained between the surface and two planes which cut the axis of x at right angles. The latter integral is to be understood as a line integral taken along the curve, and it represents the area of the portion of the curved surface which is contained between two planes at right angles to the axis of x.
(v.) When we use curvilinear coordinates ξ, η which are conjugate functions of x, y, that is to say are such that
and
the Jacobian ∂(ξ, η)/∂(x, y) can be expressed in the form
and in a number of equivalent forms. The area of any portion of the plane is represented by the double integral
where J denotes the above Jacobian, and the integration is taken through a suitable domain. When the boundary consists of portions of curves for which ξ = const., or η = const., the above is generally the simplest way of evaluating it.
(vi.) The problem of “rectifying” a plane curve, or finding its length, is solved by evaluating the integral
or, in polar coordinates, by evaluating the integral
In both cases the integrals are line integrals taken along the curve.
(vii.) When we use curvilinear coordinates ξ, η as in (v.) above, the length of any portion of a curve ξ = const. is given by the integral
taken between appropriate limits for η. There is a similar formula for the arc of a curve η = const.
(viii.) The area of a surface z = ƒ(x, y) can be expressed by the formula
When the coordinates of the points of a surface are expressed as functions of two parameters u, v, the area is expressed by the formula
When the surface is referred to three-dimensional polar coordinates r, θ, φ given by the equations
and the equation of the surface is of the form r = ƒ(θ, φ), the area is expressed by the formula
The surface integral of a function of (θ, φ) over the surface of a sphere r = const. can be expressed in the form
In every case the domain of integration must be chosen so as to include the whole surface.
(ix.) In three-dimensional polar coordinates the Jacobian
The volume integral of a function F (r, θ, φ) through the volume of a sphere r = a is
(x.) Integrations of rational functions through the volume of an ellipsoid x^{2}/a^{2} + y^{2}/b^{2} + z^{2}/c^{2} = 1 are often effected by means of a general theorem due to Lejeune Dirichlet (1839), which is as follows: when the domain of integration is that given by the inequality
where the a’s and α’s are positive, the value of the integral
is
If, however, the object aimed at is an integration through the volume of an ellipsoid it is simpler to reduce the domain of integration to that within a sphere of radius unity by the transformation x = aξ, y = bη, z = cζ, and then to perform the integration through the sphere by transforming to polar coordinates as in (ix).
56. Methods of approximate integration began to be devised very
early. Kepler’s practical measurement of the focal sectors
of ellipses (1609) was an approximate integration, as also
was the method for the quadrature of the hyperbola given
by James Gregory in the appendix to his Exercitationes
Approximate
and Mechanical Integration.
geometricae (1668). In Newton’s Methodus differentialis
(1711) the subject was taken up systematically. Newton’s
object was to effect the approximate quadrature of a given curve by
making a curve of the type
pass through the vertices of (n+1) equidistant ordinates of the given curve, and by taking the area of the new curve so determined as an approximation to the area of the given curve. In 1743 Thomas Simpson in his Mathematical Dissertations published a very convenient rule, obtained by taking the vertices of three consecutive equidistant ordinates to be points on the same parabola. The distance between the extreme ordinates corresponding to the abscissae x = a and x = b is divided into 2n equal segments by ordinates y_{1}, y_{2}, . . . y_{2n−1}, and the extreme ordinates are denoted by y_{0}, y_{2n}. The vertices of the ordinates y_{0}, y_{1}, y_{2} lie on a parabola with its axis parallel to the axis of y, so do the vertices of the ordinates y_{2}, y_{3}, y_{4}, and so on. The area is expressed approximately by the formula
which is known as Simpson’s rule. Since all simple integrals can be represented as areas such rules are applicable to approximate integration in general. For the recent developments reference may be made to the article by A. Voss in Ency. d. Math. Wiss., Bd. II., A. 2 (1899), and to a monograph by B. P. Moors, Valeur approximative d’une intégrale définie (Paris, 1905).
Many instruments have been devised for registering mechanically the areas of closed curves and the values of integrals. The best known are perhaps the “planimeter” of J. Amsler (1854) and the “integraph” of Abdank-Abakanowicz (1882).