OBTE — AATD = IBTG —AATD = ah2 tan GTB — *h2 tan DAT. Hence, if the angle which the tangent at the extremity of the ordinate uo makes with the axis of x is denoted by Ike, we have
area from uo to ul=ah(uo + 141) — 12h2(tan ''I,l — tan #o), u1 to u2=ah(u1 + 142) — ri h2(tan y'2 — tan >G,),
um1 to um ° ah(un,_1 + um)  i h2(tan ¢,n — tan 1Ym_1) ; and thence, by summation,  „ A =C1 1'sh2(tan 14, — tan >'o).
This, in the notation of §§ 46 and 54, may be written
A=C1 + [ — r'1h2u',x=xn'
Since h = H/m, the inaccuracy in taking C1 as the area varies as I /m2.
It might be shown in the same way that
A=T1+ le(tan,`m—tan,po) =T1+ [ Ah2u']z=xe'
75. The above formulae apply only to a parabolic trapezette Their generalization is given by the EulerMaclaurin formula
A= f xo udx = C1 + [  1' h2u' + 72b h4u"'  sw .lbh6uv +
vii — m •
3swiabw h9u .. x
x xo ,
and an analogous formula (which may be obtained by substituting ih and C i It and C1 in the above and then expressing Ti as zCiC1) ~'
A =f xu udx=T1+ [4h2u'6f~bh4u,,,+w~785bhe74"
T/i 2a8wh2uvii+ .. x = x,,, x=xo
To apply these, the differential coefficients have to be expressed in terms of differences.
76 If we know not only the ordinates uo, ul, . . . or ui, . . . , but also a sufficient number of the ordinates obtained by continuing the series outside the trapezette, at both extremities, we can use centraldifference formulae, which are by far the most convenient.
The formulae of § 75 give 9~
A= CI +h [—21tibu+iµbau.—zilbilb3u+se$s4ss~wbp.b7u—... ] xxo r A=TI +h [s'Abu—63bbbsu+v sabbu—as# +... ]x=x"'
xxo.
77. If we do not know values of u outside the figure, we must use advancing or receding differences. The formulae usually employed are
A=C1+h AAuo—st4A2uo+ A2uo1hA4uo+ ...
t +11EA'um— A'2um+720A'aum—ThA'4Um + ... ,
A=T1+h  Aui+214A2ui> &ui+ A4ui  .. ,
 E esaA'auma. + e so68iA'4umi ...
where A, A2, . . . have the usual meaning (Auo=u1uo, A2uo=
Au1  Auo ), and A', A'2, ... denote differences read back
wards, so that A'um = um—1 — um, A'2um =14m—2 2um_S+ um,
The calculation of the expressions in brackets may be simplified by taking the pairs in terms from the outside; i.e. by finding the successive differences of uo + um, u1 + or of ui +
ulAn alternative method, which is in some ways preferable, is to complete the table of differences by repeating the differences of the highest order that will be taken into account (see INTERPOLATION), and then to use centraldifference formulae.
78. In order to find the corrections in respect of the terms shown in square brackets in the formulae of § 75, certain ordinates other than these used for C1 or Tl are sometimes found specially. Parmentier's rule, for instance, assumes that in addition to ui, ui ... . u,,,_i, we know uo and um ; and ui  uo and um  um_i are taken
to be equal to 'shu'o and ahu'm respectively. These methods are not tc be recommended except in special cases.
79. By replacing h in § 75 by 2h, 3h, . and eliminating h2u', h'u"', ... , we obtain exact formulae corresponding to the approximate formulae of § 70. The following are the results (for the formulae involving chordal areas), given in terms of differential coefficients and of central differences. They are not so convenient as the formulae of § 76, but they serve to indicate the degree of accuracy of the approximate formulae. The expressions in square brackets are in each case to be taken as relating to the extreme values x=xo and x=xm, as in §§ 75 and 76.
(i) A=i(4C1—C2)+[— hh4u"+Ts1rsh0u to bohsuvII+ ... ]
_ (4C1—CO +1t[—T bµ63u+Tdi T 4 u—1d'.P2jojsb&u+...].
(ii) A=i(9C1—Ca)+[hh4u'" h s~bhau~_ 1 215h8u~ I . . . ]
_ i L(9C1 Ca) +h[ fib µban+a$ wµbau — z i so27bSµb7u+... ].
(iii) A=46(64Ci20C2+C4)+[wf h6u'+Thhsuvii ... ]
=; (f14C12oC2+C4)+h[s sµbsutsahwub1u . • • ].
(iv) A=A(15C6C2+Ca)+[xshhsuv rT owhsuvi+_ . . =;of15C16C2+Ca)+h[aawµbsu is1a4 esub7u ... ].
(v) A=ris(56C1—28C2+8CaC4)+[wlsaahs.uvli4.... ] =s4C(56Ci28C2+8CsC4)+ h[wi'si 7u+ ... ].
The general expression, if p, q, r, . . . are k of the factors of m, is
A = PC,,+ QCs + RC + ... + [( — )kbkh2kdx2klu +
2k+1
k+1 2k+2 74 + x — xm
( — I bk 1 dx2k+1 x = x0
where P, Q, R, . . . have the values given by the equations in § 71, and the coefficients bk, bk4.1, . . . are found from the corresponding coefficients in the ' ulerMaclaurin formula (§ 75) by multiplying them by Pp2k+Qq'+Rr21'+ ..., Pp2k+s+Qg2x+2+Rr2k+2+... ,
80. Moments of a Trapezette.—The above methods can be applied, as in §§ 59 and 6o, to finding the moments of a trapezette, when thedata are a series of ordinates. To find the pth moment, when uo, ul, u•1, . are given, we have only to find the area of a trapezette whose ordinates are xoPuo, x1Pui, x2PU2,
81. There is, however, a certain set of cases, occurring in statistics, in which the data are not a series of ordinates, but the areas Ai, Ai, . . . Am_i of the strips bounded by the consecutive ordin
ates uo, ui, ... um. The determination of the moments in these cases involves special methods, which are considered in the next two sections.
82. The most simple case is that in which the trapezette tapers out in such a way that the curve forming its top has very close contact, at its extremities, with the base; in other words, the differential coefficients u', u", u"', . are practically negligible for x=xo and for x=xm. The method adopted in these cases is to
treat the areas Ai, . as if they were ordinates placed at
the points for which x=xi, x=xi, . , to calculate the moments on this assumption, and then to apply certain corrections. If the first, second, . . . moments, so calculated, before correction are
denoted by p1, P2, . , we have
Pl = xiAi+ xiAi + ... + xm_3Am_b P2=. x2iAi + x24Ai + ... + x2m_4Am_if
Pp _ xPiAi +xPiAi+ ... + xPm_IAm _ i •
These are called the raw moments. Then, if the true moments are denoted by vl, v2,... , their values are given by
vi4iPl
V21LP2 —AMA
vanpa — 1¢h2p1
v4r 22 Pa — h2ps+3hh4Po v6=2P6 — h2Pa + h4P1
where po (or vo) is the total area Ai + Ai + . . . + Ar_i; the general expression being
vp=app—X1 p1 ih2Pp_2+X2 p!—h4PP
where _
2 (p2) • 4! (p 4) ! =111g, X2=h, '3=TSggh, 14= , ~~q a6 = Se in pp ~{~
1 . . .
The establishment of these formulae involves the use of the integral calculus.
The position of the central ordinate is given by x=si/po, and therefore is given approximately by xpi/po. To find .the moments with regard to the central ordinate, we must use this approximate value, and transform by means of the formulae given in § 32. This can be done either before cr after the above corrections are made. If the transformation is made first, and if the resulting raw moments with regard to the (approximate) central ordinate are o, 112, Ira, ... , the true moments µ1, I+z, pa, ... with regard to the central ordinate are given by
41=0
142'07r2— 1 gh2P0
44n7r4—ah2A2+ 6h4p0
A4,c rs  ah27rs
83. These results may be extended to the calculation of an expression of the form fxo u¢(x)dx, where 0(x) is a definite function of x,
and the conditions with regard to u are the same as in § 82.
(i) If ¢(x) is an explicit function of x, we have
dxo u¢(x)cIxAl (xi)+A4,G(xi)+ . • • +
where #(x) (x)2ih24,"(x)+41h4c,iv(x)—... ,
the coefficients X1, X2, .... having the values given in § 82.
(ii) If 4(x) is not given explicitly, but is tabulated for the values .. xi, xi, . . . of x, the formula of (i) applies, provided we take
Cx)=(I A82+wibb4TATbs+ ...)(k(x).
The formulae can be adapted to the case in which ck(x) is tabulated for x=xo, xi, . . .
84. In cases other than those described in § 82, the pth moment with regard to the axis of u is given by
Yp = xPmA — pS,,_i,
where A is the total area of the original trapezette, and S,,_1 is the area of a trapezette whose ordinates at successive distances h, beginning and ending with the bounding ordinates, are
o, .. xP„=i(Ai+ A;+ ... + Am]), x,P, 3A.
The value of S,_1 has to be found by a quadratureformula. The generalized formula is
fx0"'u4b(x)dx = A¢(xm) — T,
where T is the area of a trapezette whose ordinates at successive distances h are o, AO' (xi), (Ai +A4)()' (x2), . (Ai +As + . . . +
xa); the accents denoting the frst differential coefficient.
85. Volume and Moments of a Briquette.—The application of the methods of §§ 7579 to calculation of the volume of a briquette leads to complicated formulae. If the conditions are such that the methods of § 61 cannot be used, ..•r are undesirable as giving too much weight to particular ordinates, it is best to proceed in the manner indicated at the end of § 48 ; i.e. to find the areas of one set of parallel sections, and treat these as the ordinates of a trapezette whose area will be the volume of the briquette.
86. The formulae of § 82 can be extended to the case of a briquette whose top has close contact with the base all along its boundary; the data being the volumes of the minor briquettes formed by the planes x = xo, x = x1, . and y=yo, y= yi, The method of constructing the formulae is explained in § 62. If we write
Spf zu f ypx'ysu dx dy,
we first calculate the raw values ao,i, al,o, a1,1, • . of So,1, S1,o, 51,1,
. on the assumption that the volume of each minor briquette is concentrated along its midordinate (§ 44), and we then obtain the formulae of correction by multiplying the formulae of § 82 in pairs. Thus we find (e.g.)
S1,141a1a
S2,i 02,1—Ah2oo,1 S1,22o1,2— k2al,o
k2a2,o —A h2oo,2I 1}4 h2k2ao,o
53,11 a3r1 — 4h2a1,1
S3,2nas,2—ah2a1j1k2as,o+ h2k2ol,o
where ao,o is the total volume of the briquette.
87. If the data of the briquette are, as in § 86, the volumes of the minor briquettes, but the condition as to close contact is not satisfied, we have
7o f x'yqu dx dy = K + L + R — x+r,,,ynao,0
where Kmx Xgth moment with regard to plane y=o, L=y,l Xpth moment with regard to plane x=o,
and R is the volume of a briquette whose ordinate at (x.,y,) is found by multiplying by pQ x,P' y.q' the volume of that portion of the original briquette which lies between the planes x=xo, x=x., y =Yo, y = y,• The ordinates of this new briquette at the points of intersection of x=xo, x=x1, . . . with y=yo, y=yi, . are obtained from the data by summation and multiplication; and the ordinary methods then apply for calculation of its volume. Either or both of the expressions K and L will have to be calculated by means of the formula of § 84; if this is applied to both expressions, we have a formula which may be written in a more general form
f f u¢(x, y)dx dy = rig u dx dy . 4(b, q)
_Ib f'fqudxdy do(xx ,O
a a a d
— f q fbf" u dx dy do(b Y) dy
a p dY
..h f ° f q f T u dx dy d (xdy, y) dx dy.
dx
The second and third expressions on the righthand side represent areas of trapezettes, which can be calculated from the data; and the fourth expression represents the volume of a briquette, to be calculated in the same way as R above.
88. Cases of Failure.—When the sequence of differences is not such as to enable any of the foregoing methods to be applied, it is sometimes possible to amplify the data by measurement of intermediate ordinates, and then apply a suitable method to the amplified series.
There is, however, a certain class of cases in which no subdivision of intervals will produce a good result ; viz. cases in which the top of the figure is, at one extremity (or one part of its boundary), at right angles to the base. The EulerMaclaurin formula (§ 75) assumes that the bounding values of u', um, . . . are not infinite; this condition is not satisfied in the cases here considered. It is also clearly impossible to express u as an algebraical function of x and y if some value of du/dx or du/dy is to be infinite.
No completely satisfactory methods have been devised for dealing
with these cases. One method is to construct a table for interpola
tion of x in terms of u, and from this table
to calculate values of x corresponding to
E values of u, proceeding by equal intervals;
a quadratureformula can then be applied.
Suppose, for instance, that we require the
area of the trapezette ABL in fig. I I ; the
curve being at right angles to the base AL
L
at A. If QD is the bounding ordinate of
one of the component strips, we can calculate
the area of QDBL in the ordinary way. The
data for the area ADQ are a series of values of u corresponding to
equidifferent values of x; if we denote by y the distance of a point
on the arc AD from QD, we can from the series of values of u construct a series of values of y corresponding to equidifferent values of u, and thus find the area of ADQ, treating QD as the base. The process, however, is troublesome.
89. Examples of Applications.—The following are some examples of cases in which the above methods may be applied to the calculation of areas and integrals.
(i) Construction of Mathematical Tables.—Even where u is an explicit function of x, so thatfxudx may be expressed in terms of x, it is often more convenient, for construction of a table of values of such an integral, to use finitedifference formulae. The formula Of § 76 may (see DIFFERENCES, CALCULUS OF) be written
f udx=h.,uau+h_(—Aµbu+ 21 6µb3u )
= (hu—12 Phu + 64hu — . . .),
sea(hu+Ab2hu—s++'sabyhu+...).
The second of these is usually the more convenient Thus, to
construct a table of values of f'udx by intervals of h in x, we first
form a table of values of hu for the intermediate values of x; from
this obtain a table of values of (1+2'4b2_sltao84+. ..) hu tor
these values of x, and then construct the table of fxudx by successive additions. Attention must be given to the possible accumulation of errors due to the small errors in the values of u. Each of the above formulae involves an arbitrary epnstant; but this. disappears when we start the additions from a known value. of
udx.
The process may be repeated. Thus wehave
f 'f udx dx (a+b s+sasS3+ . . .)2h2u
_ (a2+ z—Tieb2+r Nut b4—sgMTbo+ ...)h2u a2(h2u + 112 b2h2u — 1h. b4h2u + • ..).
Here there are two arbitrary constants, which may be adjusted in various ways.
The formulae may be used for extending the accuracy of tables, in cases where, if v represents the quantity tabulated, hdv/dx or, h2d2v/dx2 can be conveniently expressed in terms of v and x to a greater degree of accuracy than it could be found from the table. The process practically consists in using the table as it stands for improving the first or second differences of v and then building up the table afresh.
(ii) Life Insurance.—The use of quadratureformulae is important in actuarial work, where the fundamental tables are based on experience, and the formulae applying these tables involve the use of the tabulated values and their differences.
90. The following are instances of the application of approximative formulae to the calculation of the'volumes of solids.
(i) Timber Measure.—To find the quantity of timber in a trunk with parallel ends, the areas of a few sections must be calculated as accurately as possible, and a formula applied. As the measurements can only be rough, the trapezoidal rule is the most appropriate in ordinary cases.
(ii) Gauging.—To measure the volume of a cask, it may be assumed that the interior is approximately a portion of a spheroidal figure. The formula applied can then be either Sim sons rule or a rule based on Gauss's theorem for two ordinates (§56). In the latter case the two sections are taken at distances = 1H/s/ 3 = = •2887H from the middle section, where H is the total internal length; and their arithmetic mean is taken to be the mean section of the cask. Allowance must of course be made for the thickness of the wood.
91. Certain approximate formulae for the length of an arc of a circle are obtained by methods similar to those of §§ 71 and 79. Let a be the radius of a circle, and B (circular measure) the unknown angle subtended by an arc. Then, if we divide 8 into m equal parts, and L1 denotes the sum of the corresponding. chords, so that L1=2ma sin (6/2m), the true length of the arc is
L3 + aO `~ ~ + . . . , where ¢ =B/2m. Similarly, if Lo repre3• 5•
sents the sum of the chords when m (assumed even) is replaced by im, we have an expression involving L2 and 2a,. The method of § 71 then shows that, by taking i(4L1—L2) as the value of the arc, we get rid of terms in 02. If we use cl to represent the chord 01 the whole arc, cz the chord of half the arc, and c4 the chord of ones quarter of the arc, then corresponding to (i) and (iii) of § 70 or § 79 we have (8c2 —cl) and A (256c4—4oc2+ci) as approximations to the length of the arc. The first of these is Huygens's rule.
f'udx =h.au+h( Su—g7eFS3u+. . .)
(1903). For examples of measurement of areas by geometrical construction, see G. C. Turner, Graphics applied to Arithmetic, Mensuration and Statics (1907). Discussions of the approximate calculation of definite integrals will be found in works on the infinitesimal calculus; see e.g. E. Goursat, A Course in Mathematical Analysis (1905; trans. by E. R. Hedrick). For the methods involving finite differences, see references under DIFFERENCES, CALCULUS OF; and INTERPOLATION. On calculation of moments of graphs, see W. P. Elderton, FrequencyCurves and Correlation (1906); as to the formulae of § 82, see also Biomedrika, v. 450. For mechanical methods of calculating areas and moments see CALCULATING MACHINES. (W. F. Sn.)
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