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OBTE — AATD = IBTG —AATD = ah2 tan GTB — *h2 tan DAT. Hence, if the See also:angle which the tangent at the extremity of the See also:ordinate uo makes with the See also:axis of x is denoted by Ike, we have See also: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,), um-1 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 See also:Euler-See also:Maclaurin See also: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 zCi-C1) ~' A =f xu udx=T1+ [-4h2u'-6f~bh4u,,,+w~785bhe74"- T/i 2a8wh2uvii+ .. x = x,,, x=xo To apply these, the See also:differential coefficients have to be expressed in terms of See also: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 See also:series outside the trapezette, at both extremities, we can use central-difference formulae, which are by far the most convenient. The formulae of § 75 give 9~ A= CI +h [—21tibu+iµbau.—zilbilb3u+se$s4ss~wbp.b7u—... ] x-xo r A=TI +h [s'See also:Abu—63bbbsu+v sabbu—as# +... ]x=x"' x-xo. 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+ A2uo-1hA4uo+ ... t +11EA'um— A'2um+720A'aum—ThA'4Um + ... , A=T1+h - Aui+214A2ui-> &ui+ A4ui - .. , - E esaA'aum-a. + e so68iA'4um-i ... where A, See also:A2, . . . have the usual meaning (Auo=u1-uo, 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 + See also:ulAn alternative method, which is in some ways preferable, is to See also:complete the table of differences by repeating the differences of the highest See also:order that will be taken into See also:account (see See also:INTERPOLATION), and then to use central-difference 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 See also: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 See also: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 See also:case to be taken as See also:relating to the extreme values x=xo and x=xm, as in §§ 75 and 76. (i) A=i(4C1—C2)+[— hh4u"+Ts1rsh0u to b-ohsuvII+ ... ] _ (4C1—CO +1t[—T bµ63u+Tdi T 4 u—1d'.P2jojsb&u+...]. (ii) A=i(9C1—Ca)+[--hh4u'" h s~bhau~_ 1 21-5h8u~ -I- . . . ] _ i L(9C1- Ca) +h[ -fib µSee also:ban+a$ wµbau — z i so27bSµb7u+... ]. (iii) A=46(64Ci-20C2+C4)+[-wf h6u'+Thhsuvii- ... ] =; (f14C1-2oC2+C4)+h[-s sµbsutsah-wub1u- . • • ]. (iv) A=A(15C-6C2+Ca)+[-xshhsuv rT owhsuvi+_ . . =;of15C1-6C2+Ca)+h[-aawµbsu i-s1a4 esub7u- ... ]. (v) A=ris(56C1—28C2+8Ca-C4)+[-wlsa-ahs.uvli4.... ] =s4C(56Ci-28C2+8Cs-C4)+ h[-wi'si 7u+ ... ]. The See also:general expression, if p, q, r, . . . are k of the factors of m, is A = PC,,+ QCs + See also: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 '- uler-Maclaurin 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 See also: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 See also:simple case is that in which the trapezette tapers out in such a way that the See also:curve forming its See also:top has very See also:close contact, at its extremities, with the See also: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 See also: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 V2-1-L-P2 —AMA See also:van-pa — 1¢h2p1 v4r 22 Pa — h2ps+3hh4Po v6=2P6 — h2Pa + h4P1 where po (or vo) is the See also:total area Ai + Ai + . . . + Ar_i; the general expression being vp=app—X1 p1 ih2Pp_2+X2 p!—h4PP where _ 2 (p-2) • 4! (p -4) ! =111g, X2=h, '3=TSggh, 14= , ~~q a6 = Se in pp ~{~ 1 . . . The See also: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 x-pi/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=0 142'0-7r2— 1 gh2P0 44-n-7r4—ah2A2+ 6h4p0 See also:A4,c -rs -- ah27rs 83. These results may be extended to the calculation of an expression of the See also:form fxo u¢(x)dx, where 0(x) is a definite See also: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)cIx--Al (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+wibb4-TATbs+ ...)(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 See also: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 See also:quadrature-formula. 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. See also:Volume and Moments of a See also:Briquette.—The application of the methods of §§ 75-79 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 See also: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 See also: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 See also:mid-ordinate (§ 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,2-I 1}4 h2k2ao,o 53,11 -a3r1 — 4h2a1,1 S3,2-n-as,2—ah2a1--j1k2as,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 See also: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 See also: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 See also: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 right-See also:hand See also:side represent areas of trapezettes, which can be calculated from the data; and the See also: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 some-times 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 See also:good result ; viz. cases in which the top of the figure is, at one extremity (or one See also:part of its boundary), at right angles to the base. The Euler-Maclaurin formula (§ 75) assumes that the bounding values of u', um, . . . are not See also:infinite; this condition is not satisfied in the cases here considered. It is also clearly impossible to See also: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 quadrature-formula 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 See also: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 finite-difference formulae. The formula Of § 76 may (see DIFFERENCES, CALCULUS OF) be written f udx=h.,uau+h_(—Aµbu+ 21 6µb3u ) = (hu—12 Phu + 64hu — . . .), See also: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. See also:Attention must be given to the possible See also: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 we-have f 'f udx dx (a+b s+sasS3+ . . .)2h2u _ (a2+ z—Tieb2+r See also: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 See also:building up the table afresh. (ii) See also:Life See also:Insurance.—The use of quadrature-formulae is important in actuarial See also: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) See also:Timber Measure.—To find the quantity of timber in a See also: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 See also:Gauss's theorem for two ordinates (§56). In the latter case the two sections are taken at distances = 1H/s/ 3 = = •2887H from the See also:middle See also:section, where H is the total See also:internal length; and their See also:arithmetic mean is taken to be the mean section of the cask. See also:Allowance must of course be made for the thickness of the See also: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 See also:radius of a circle, and B (circular measure) the unknown angle subtended by an arc. Then, if we See also:divide 8 into m equal parts, and L1 denotes the sum of the corresponding. chords, so that L1=2ma See also:sin (6/2m), the true length of the arc is L3 + aO `~ ~ + . . . , where ¢ =B/2m. Similarly, if Lo repre-3• 5• sents the sum of the chords when m (assumed even) is replaced by See also: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 See also:half the arc, and c4 the chord of ones See also: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 See also:Huygens's rule. f'udx =h.au+h( Su—g7eFS3u+. . .) (1903). For examples of measurement of areas by geometrical construction, see G. C. See also:Turner, Graphics applied to Arithmetic, See also:Mensuration and See also:Statics (1907). Discussions of the approximate calculation of definite integrals will be found in See also:works on the infinitesimal calculus; see e.g. E. Goursat, A Course in Mathematical See also: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, Frequency-Curves and Correlation (1906); as to the formulae of § 82, see also Biomedrika, v. 450. For See also:mechanical methods of calculating areas and moments see CALCULATING See also:MACHINES. (W. F. Additional information and CommentsThere are no comments yet for this article.
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