Part 21

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 _ / 1 | (1 - x²)^-½ log x dx = -½ [pi] log 2, _/ 0

although the indefinite integral of (1 - x²)^-½ 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 _ / [oo] x^(a-1) [pi] (i.) | ------- dx = ---------, (1 > a > 0), _/ 0 1 + x sin a[pi]

_ / [oo] x^(a-1) - x^(b-1) (ii.) | ----------------- dx = [pi](cot a[pi] - cot b[pi]), (0 < a or b < 1), _/ 0 1 - x

_ / [oo] x^(a-1) log x [pi]² (iii.) | ------------ dx = ----------, (a > 1), _/ 0 x - 1 sin² a[pi]

_ / [oo] (iv.) | x²·cos 2x·e^-x² dx = -¼ e^-1 [root][pi], _/ 0

_ / 1 1 - x² dx [pi] (v.) | ------- ----- = log tan ----, _/ 0 1 + x^4 log x 8

_ / [oo] sin mx / 1 1 1 \ (vi.) | -------------- dx = ½ ( ------- - --- + --- ), _/ 0 e^(2[pi]x) - 1 \ e^m - 1 m 2 /

_ / [pi] (vii.) | log(1 - 2[alpha] cos x + [alpha]²) dx = 0 _/ 0

or 2[pi]log [alpha] according as [alpha] < or > 1, _ / [oo] sin x (viii.) | ----- dx = ½[pi], _/ 0 x

_ / [oo] cos ax (ix.) | ------- dx = ½[pi]b^-1 e^(-ab), _/ 0 x² + b²

_ / [oo] cos ax - cos bx (x.) | --------------- dx = ½[pi](b - a), _/ 0 x²

_ / [oo] cos ax - cos bx b (xi.) | --------------- dx = log ---, _/ 0 x a

_ / [oo] cos x - e ^(-mx) (xii.) | ---------------- dx = log m, _/ 0 x

_ / [oo] (xiii.) | e^(-x²+2ax) dx = [root][pi].e^(a2), _/ -[oo]

_ _ / [oo] / [oo] (xiv.) | x^-½ sin x dx = | x^-½ cos x dx = [root](½[pi]), _/ 0 _/ 0

Multiple Integrals.

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 [f](x, y) over an area; in the case of three variables x, y, z we integrate a function [f](x, y, z) through a volume. The integral of a function [f](x, y) over an area in the plane of (x, y) is denoted by _ _ / / | | [f](x, y) dx dy. _/_/

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 [f] 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 [f](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 _ _ / b { / [f]2(x) } | dx { | [f](x, y) dy }, _/ a { _/ [f]1(x) }

and is called a "repeated integral." The identification of a surface integral, such as [int][int][f](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 _ _ _ / / / | | | [f](x, y, z) dx dy dz. _/_/_/

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."

Surface Integrals.

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 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 [f] 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 _ _ / / | | [f] dS. _/_/

Line Integrals.

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 written _ / | [f] ds _/

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 _ _ / / dx dy dz \ / | ( u -- + v -- + w -- ) ds or | (u dx + v dy + w dz). _/ \ ds ds ds / _/ s

In like manner surface integrals usually present themselves in the form _ _ / / | | (l[xi] + m[eta] + n[zeta]) dS _/_/

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 _ / ½ | (x dy - y dx), _/

or as _ _ / / | | dx dy, _/_/

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.

Theorems of Green and Stokes.

53_a_. 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 equation _ _ _ _ _ / / / / ð[xi] ð[eta] ð[zeta]\ / / | | | ( ----- + ------ + ------- )dx dy dz = | | (l[xi] + m[eta] + n[zeta]) dS, _/_/_/ \ ðx ðy ðz / _/_/

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., _ _ _ / / / ð[xi] ð[eta]\ / / dy dx \ | | ( ----- + ------ ) dx dy = | ( [xi] -- - [eta] -- ) ds, _/_/ \ ðx ðy / _/ \ ds ds /

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 _ _ _ / / / { /ðw ðv\ /ðu ðw\ /ðv ðu\ } | (u dx + v dy + w dz) = | | { l( -- - -- ) + m( -- - -- ) + n( -- - -- ) }dS, _/ s _/_/ { \ðy ðz/ \ðz ðx/ \ðx ðy/ }

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 ð[xi]/ðx, ð[eta]/ðy, ð[zeta]/ðz 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.

Change of Variables in a Multiple Integral.

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 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 x1, x2, ..., x_n, and those of the other set by u1, u2, ..., u_n, we have the relation

ð(x1, x2, ..., x_n) dx1 dx2 ...dx_n = ------------------- du1 du2 ... du_n. ð(u1, u2, ..., u_n)

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 [f](x, y) over the area within a circle given by x² + y² = a², and we introduce polar coordinates so that x = r cos [theta], y = r sin [theta], we find that r is the value of the Jacobian, and that all points within or on the circle are given by a [>=] r [>=] o, 2[pi][>=][theta][>=]o, and we have _ _ _ _ / a / [root](a²-x²) / a /2[pi] | dx | [f](x, y) dy = | dr | f(r cos [theta], r sin [theta]) r d[theta]. _/-a _/-[root](a²-x²) _/ 0 _/ 0

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:-- _ _ _ _ /a / b /tan^-1 b/a /a sec [theta] | dx | [f](x, y) dy = | d[theta] | [f](r cos [theta], r sin [theta]) r dr _/0 _/ 0 _/ 0 _/0 _ _ / ½[pi] /b cosec [theta] + | d[theta] | [f](r cos [theta], r sin [theta]) r dr. _/tan^-1 b/a _/ 0

55. A few additional results in relation to line integrals and multiple integrals are set down here.

Line Integrals and Multiple Integrals.

(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 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 _ _ / / | ½ r² d[theta] or | ½ p ds, _/ _/

where r, [theta] 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 _ / [pi] | y² dx, _/

and the area of the surface is expressed by the formula _ / 2[pi] | y ds, _/

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 [xi], [eta] which are conjugate functions of x, y, that is to say are such that

ð[xi]/ðx = ð[eta]/ðy and ð[xi]/ðy = -ð[eta]/ðx,

the Jacobian ð([xi], [eta])/ð(x, v) can be expressed in the form

/ð[xi]\² /ð[eta]\² ( ----- ) + ( ------ ), \ ðx / \ ðx /

and in a number of equivalent forms. The area of any portion of the plane is represented by the double integral _ _ / / | | J^-1 d[xi] d[eta], _/_/

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 [xi] = const., or [eta] = 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 _ / { /dy\² }½ | { 1 + ( -- ) } dx, _/ { \dx/ }

or, in polar coordinates, by evaluating the integral _ / { / dr \² }½ | { r² + ( -------- ) } d[theta]. _/ { \d[theta]/ }

In both cases the integrals are line integrals taken along the curve.

(vii.) When we use curvilinear coordinates [xi], [eta] as in (v.) above, the length of any portion of a curve [xi] = const. is given by the integral _ / | J^-½ d[eta] _/

taken between appropriate limits for [eta]. There is a similar formula for the arc of a curve [eta] = const.

(viii.) The area of a surface z = [f](x, y) can be expressed by the formula _ _ / / { /ðz\² /ðz\² }½ | | { 1 + ( -- ) + ( -- ) } dx dy. _/_/ { \ðx/ \ðy/ }

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 _ _ _ _ / / | { ð(y, z) }² { ð(z, x) }² { ð(x, y) }² |½ | | | { ------- } + { ------- } + { ------- } | du dv. _/_/ |_ { ð(u, v) } { ð(u, v) } { ð(u, v) } _|

When the surface is referred to three-dimensional polar coordinates r, [theta], [phi] given by the equations

x = r sin [theta] cos [phi], y = r sin [theta] sin [phi], z = r cos [theta],

and the equation of the surface is of the form r = [f]([theta], [phi]), the area is expressed by the formula _ _ _ _ / / | { / ðr \² } / ðr \² |½ | | r | { r² + ( -------- ) } sin² [theta] + ( ------ ) | d[theta] d[phi]. _/_/ |_ { \ð[theta]/ } \ð[phi]/ _|

The surface integral of a function of ([theta], [phi]) over the surface of a sphere r = const. can be expressed in the form

_ _ /2[pi] /[pi] | d[phi] | F([theta], [phi]) r² sin [theta] d[theta]. _/ 0 _/ 0

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

ð(x, y, z) -------------------- = r² sin [theta] ð(r, [theta], [phi])

The volume integral of a function F (r, [theta], [phi]) through the volume of a sphere r = a is _ _ _ / a /2[pi] /[pi] | dr | d[phi] | F(r, [theta], [phi]) r² sin [theta] d[theta]. _/ 0 _/ 0 _/ 0

(x.) Integrations of rational functions through the volume of an ellipsoid x²/a² + y²/b² + z²/c² = 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

/x1\[alpha]1 /x2\^[alpha]2 /x_n\[alpha]_n ( -- ) + ( -- ) + ... + ( --- ) [<=] 1 \a1/ \a2/ \a_n/

where the a's and [alpha]'s are positive, the value of the integral _ _ / / | | ... x1^(n1-1)·x2^(n2-1) ... dx1 dx2 ... _/_/

a1^(n1) a2^(n2) ... [Gamma] (n1/[alpha]1) [Gamma] (n2/[alpha]2) is --------------------- ---------------------------------------------. [alpha]1 [alpha]2 ... [Gamma](1 + n1/[alpha]1 + n2/[alpha]2 + ... )

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 = a[xi], y = b[eta], z = c[zeta], and then to perform the integration through the sphere by transforming to polar coordinates as in (ix).

Approximate and Mechanical Integration.

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 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

y = a0 + a1x + a2x² + ... + a_n x^n

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 y1, y2, ... y(2n-1), and the extreme ordinates are denoted by y0, y(2n). The vertices of the ordinates y0, y1, y2 lie on a parabola with its axis parallel to the axis of y, so do the vertices of the ordinates y2, y3, y4, and so on. The area is expressed approximately by the formula

{(b - a)/6n} [y0 + y_(2n) + 2 (y2 + y4 + ... + y_(2n-2)) + 4(y1 + y3 + ... + y_(2n-1)],

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).