Page:EB1911 - Volume 14.djvu/582

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 x²+y² = a², 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

$\int _{-a}^{a}dx\int _{-{\sqrt {}}a^{2}-x^{2}}^{{\sqrt {}}a^{2}-x^{2}}f(x,y)dy=\int _{0}^{a}dr\int _{0}^{2\pi }f(r\cos \theta ,r\sin \theta )rd\theta .$

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:—

$\int _{0}^{a}dx\int _{0}^{b}f(x,y)dy=\int _{0}^{\tan ^{-1}b/a}d\theta \int _{0}^{a\sec x}f(r\cos \theta ,r\sin \theta )rdr$

$+\int _{\tan ^{-1}b/a}^{{\frac {1}{2}}\pi }d\theta \int _{0}^{b\,cosec\,\theta }f(r\cos \theta ,r\sin \theta )rdr$

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

$\int {\tfrac {1}{2}}r^{2}d\theta$  or$\int {\tfrac {1}{2}}pds$

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

$\pi \int y^{2}dx$

and the area of the surface is expressed by the formula

$2\pi \int yds$

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

$\partial \xi /\partial x=\partial \eta /\partial y$ and $\partial \xi /\partial y=\partial \eta /\partial x,$

the Jacobian ∂(ξ, η)/∂(x, y) can be expressed in the form

$\left({\frac {\partial \xi }{\partial x}}\right)^{2}+\left({\frac {\partial \eta }{\partial y}}\right)^{2},$

and in a number of equivalent forms. The area of any portion of the plane is represented by the double integral

$\iint J^{-1}d\xi d\eta ,$

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 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

$\int \left\{1+\left({\frac {dy}{dx}}\right)^{2}\right\}^{\frac {1}{2}}dx,$

or, in polar coordinates, by evaluating the integral

$\int \left\{r^{2}+\left({\frac {dr}{d\theta }}\right)^{2}\right\}^{\frac {1}{2}}d\theta ,$

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

$\int J^{-{\tfrac {1}{2}}}d\eta$

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

$\iint \left\{1+\left({\frac {\partial z}{\partial x}}\right)^{2}+\left({\frac {\partial z}{\partial y}}\right)^{2}\right\}^{\frac {1}{2}}dxdy.$

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

$\iint \left\lbrack \left\{{\frac {\partial (y,z)}{\partial (u,v)}}\right\}^{2}+\left\{{\frac {\partial (z,x)}{\partial (u,v)}}\right\}^{2}+\left\{{\frac {\partial (x,y)}{\partial (u,v)}}\right\}^{2}\right\rbrack ^{\frac {1}{2}}dudv.$

When the surface is referred to three-dimensional polar coordinates r, θ, φ 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 = ƒ(θ, φ), the area is expressed by the formula

$\iint r\left\lbrack \left\{r^{2}+\left({\frac {\partial r}{\partial \theta }}\right)^{2}\right\}\sin ^{2}\theta +\left({\frac {\partial r}{\partial \phi }}\right)^{2}\right\rbrack ^{\frac {1}{2}}d\theta d\phi .$

The surface integral of a function of (θ, φ) over the surface of a sphere r = const. can be expressed in the form

$\int _{0}^{2\pi }d\phi \int _{0}^{\pi }F(\theta ,\phi )r^{2}\sin \theta d\theta .$

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

${\frac {\partial (x,y,z)}{\partial (r,\theta ,\phi )}}=r^{2}\sin \theta .$

The volume integral of a function F (r, θ, φ) through the volume of a sphere r = a is

$\int _{0}^{a}dr\int _{0}^{2\pi }d\phi \int _{0}^{\pi }F(r,\theta ,\phi )r^{2}\sin \theta d\theta .$

(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

$\left({\frac {x_{1}}{a_{1}}}\right)^{a_{1}}+\left({\frac {x_{2}}{a_{2}}}\right)^{a_{2}}+\ldots +\left({\frac {x_{n}}{a_{n}}}\right)^{a_{n}}\leq 1,$

where the a’s and α’s are positive, the value of the integral

$\iint \ldots x_{1}^{n_{1}-1}\cdot x_{2}^{n_{2}-1}\ldots dx_{1}dx_{2}\ldots$

is

${\frac {a_{1}^{n_{2}}a_{2}^{n_{2}}\ldots }{a_{1}a_{2}\ldots }}{\frac {\Gamma \left({\frac {n_{1}}{a_{1}}}\right)\Gamma \left({\frac {n_{2}}{a_{2}}}\right)}{\Gamma \left(1+{\frac {n_{1}}{a_{1}}}+{\frac {n_{2}}{a_{2}}}\right)}}.$

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

$y=a_{0}+a_{l}x+a_{2}x^{2}+\ldots +a_{n}x^{n}$

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

$\left\{(b-a)/6n\right\}\left\lbrack y_{0}+y_{2n}+2(y_{2}+y_{4}+\ldots +y_{2n-2})\right.\left.+4(y_{1}+y_{3}+\ldots +y_{2n-1})\right\rbrack ,$

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).

Bibliography.—For historical questions relating to the subject the chief authority is M. Cantor, Geschichte d. Mathematik (3 Bde., Leipzig, 1894–1901). For particular matters, or special periods, the following may be mentioned: H. G. Zeuthen, Geschichte d. Math. im Altertum u. Mittelalter (Copenhagen, 1896) and Gesch. d. Math. im XVI. u. XVII. Jahrhundert (Leipzig, 1903); S. Horsley, Isaaci Newtoni opera quae exstant omnia (5 vols., London, 1779–1785); C. I. Gerhardt, Leibnizens math. Schriften (7 Bde., Leipzig, 1849–1863); Joh. Bernoulli, Opera omnia (4 Bde., Lausanne and Geneva, 1742). Other writings of importance in the history of the subject