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 domam
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+y“=a2, and we introduce polar
coordinates so that x =r cos 6, y=r sin 0, we find that f is the value
of the jacobian, and that all points within or on the circle are given
by ag rg 0, 27l'>9'§ 0, and we have

a a2-x2 a 21r,

f adxf1/j<a2;)f(x, y)dy=f0dff0f(rcos0, rsm 6)rd0. If we have to integrate over the area of a rectangleaé xé 0, bg ya 0, and we transform to polar coordinates, the integral becomes the sum of two integrals, as follows:-

f;dx]; f(x, y)dy= ganulb/ad0; Sec 9f(1 cos0, rsin 0)rdr 1

+f£'; 1b/$19 Scoscc of (1 cos0, rsin 0)rdr. fl

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 5x:gr us variables is made, the limits of integration with respect md to the new variable must be such that the domain of M M le integration is the same as before. This condition may n';e;us require the replacing of the original integral by the sum of two or more simple integrals.

(ii.) The line integral of a erfect 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

f%r2d0 or fépds,

where r, 0 are polar coordinates, and p is the perpendicular drawn from a fixed point to the tangent. The integrals are to he 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 radn 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 vrfyzdx,

and the area of the surface is expressed by the formula zwfyds,

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 E, 1; which are conjugate functions of x, y, that is to say are such that 65/dx = 61|/dy and 62/dy =-61;/dx,

the Jacobian 6(£, 11)/6(x, y) can be expressed in the form dx 6x

(gi) 2+ (QV

and in a number of equivalent forms. The area of any portion of the plane is represented by the double integral ff l“'d£dm

where ] denotes the above jgcobian, and the integration is taken through a suitable domain. hen the boundary consists of portions of curves for which £=const., or 11 =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 i

faI+<<f-fw»

dx

or, in polar coordinates, by evaluating the integral f§ »+<@>”-@

dt) 5

ln both cases the integrals are line integrals taken along the curve. (vii.) When we use curvilinear coordinates 2, 11 as in (v.) above, the length of any portion of a curve E=const. is given by the integral fJ'¥dn

taken between appropriate limits for -q. There is a similar formula for the arc of a curve 1|=const.

(viii.) The area of a surface z-=f(x, y) can be expressed by the formula

ff%1+<%§)'+

When .the coordinates of the points of a surface are expressed as functions of two parameters u, v,

§ 6(y, z) $2 § 6(z, x) 2

ff l e a"<~, to + aft. ti

When the surface is referred to

r, 0, ¢ given by the equations

x=r sin6 cos 4>, y=r sin0 sin 41, z=rcos0, S306 y) 2

the area is expressed by the formula

- T

l' I a(u' U); ji

three-dimensional polar coordinates

and the equation of the surface is of the form r=f(0, ¢), the area is expressed by the formula

fffli s> 21 ~~
~~

~~The surface integral of a function of (0, ¢>) over the surface of a sphere
r=-const. can be expressed in the form
f;"d4>f;F<e, ¢>f2Sinade
~~

~~In every case the domain of integration must be chosen
include the whole surface.
~~

~~SO as to
~~

~~(ix.) In three-dimensional polar coordinates the jacobian
0(x.y.2) -
~~

~~6016, ¢)-r2s1nl9.
~~

~~The volume integral of a function F (r, 6, da) through the volume of a
sphere r=a is
~~

~~I f; drfgnddxfig F(f, 0, ¢)r2 sin 0d0.
(x.) Integrations of rational functions through the volume of an
ellipsoid x2/a' -l-y"/bz +22/4:2 = 1 are often eHected 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
22 “1 52 “2 & an
~~

~~(111) + (112) + +<'1f-> é I
~~

~~where the a's and a's are positive, the value of the integral
ff.. xml". x2”z'1 . ..dx1dxg . . .
~~

~~is a, "1a2"2... F<a1 F<a¢>'
~~

~~“1“2'T'I' 1+' ¥+@+...
~~

~~U-1 (12
~~

~~If, however, the object aimed at is an integration through the volume
of an ellipsoid it is sim ler to reduce the domain of integration to
that within a sphere ofp radius unity by'the transformation x=a£,
y=b1;, z=c§ ', and then to erform the integration through the
sphere by transforming to polar coordinates as in (ix).
56. Methods of approximate integration began to be dev
early. Kepler's practical measurement of the ocal sectors
ised very
~~

~~of ellipses (1609) was an ap roximate integration, as also Apfimxhd
was the method for the quadrature of the hyperbola given fn" Zn;
by James Gregory in the appendix to his Exercitationes TI an
geometrical (1668). In Newton's Methodus dijerentialis ca "
(171 1) the subject was taken up systematically. N ewton's t"°3°"'“°"
object was to effect the approximate quadrature of a given
making a curve of the type
~~

~~y =a0--a1x+a2x'+ . . . +a, ., x"
~~

~~curve by
~~

~~pass through the vertices of (n-I-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 yl, yg, . . . y¢, .;,
and the extreme ordinates are denoted by yo, y2, .. The vertices of
the ordinates yo, yi, yg lie on a parabola with its axis parallel to the
axis of y, so do the vertices of the ordinates y2, ys, y4, and so on.
The area is expressed approximately by the formula
{(71-ft)/61»}[y<>+:>'2»+2(y=+y1+ ~~ - +y2»-2) +4(y1+y=+ »— +;v2, ..l)l.
which is known as Simpson's rule. Since all simple integrals can be
represented as areas such rules are applicable to a proximate 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 déjinie (Paris, 1905).,
Many instruments have been devised for re istering 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).
BIBLIOG RA PHY.—F or historical questions relating to the subject th
chief authority is M. Cantor, Geschichte d. Mathemattk (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 X VI. u. X VII. .fahrhundert (Leipzig, 1903); S. Horsley, Isaaci
Newtoni opera quae exstant omnia (5 vols., London, 1779-1785);
C. I. Gerhardt, Leibnizens math. Schngen (7 Bde., Leipzig, 1849-1863);
]oh. Bernoulli, Opera omnia (4 de., Lausanne and Geneva,
~~

~~
1742). Other writings of importance in the history of the subject~~