Elements of analysis as applied to the mechanics of engineering and machinery . i — !/o) ^2 — (^2 — !/o) ^l) ^ ^ ^ 6 ^, r^„ — X,) J ^ y. ^, 2 V. 6^,(^2 — ^,) If we had ^^ ~ = —, we should have to consider a rectilinear yi — yo^i boundary, and there would then be, simply, as also, ^ __2V±J/2 If, further, we had merely x^ = 2 x^, therefore, y^ equally distantfrom the boundary ordinates y^^ and y^^ there would be [Art. 38. ELEMENTS OF ANALYSIS. 63 cc„ ?^= (!/o + 4 2/1 + yd -^- (vid. Art. 30), and „^ ^2/0 + 47/, + y. If an area M^M^P.^P^, Eig. 50,is determined by four co-ordinates M^ P„ = 2/0, ^^1
Elements of analysis as applied to the mechanics of engineering and machinery . i — !/o) ^2 — (^2 — !/o) ^l) ^ ^ ^ 6 ^, r^„ — X,) J ^ y. ^, 2 V. 6^,(^2 — ^,) If we had ^^ ~ = —, we should have to consider a rectilinear yi — yo^i boundary, and there would then be, simply, as also, ^ __2V±J/2 If, further, we had merely x^ = 2 x^, therefore, y^ equally distantfrom the boundary ordinates y^^ and y^^ there would be [Art. 38. ELEMENTS OF ANALYSIS. 63 cc„ ?^= (!/o + 4 2/1 + yd -^- (vid. Art. 30), and „^ ^2/0 + 47/, + y. If an area M^M^P.^P^, Eig. 50,is determined by four co-ordinates M^ P„ = 2/0, ^^1 P^ = y,, ^. P. =2/.,M^P^ = y^^ which are at equal dis-tances from each other, the mag-nitude of the same may be deter-mined approximately in the follow-ing manner. If we represent the line of baseM^ Jig by ^3, and three ordinatesN^Q^^ N^Q.^-, N.^Qsf inserted betweeny^ and y^ at equal distances fromeach qther, by z^^ z^, z,,, we can ]putthe surface approximately; ^o^s^a^o = -^= (i 2/0 + ^0 + ^x + ^2 + i y-s) ^~.But we have now, L 9,7. J^ 9,^ \-%. Nj^Mi K^ M3N3 ^1 + ^2 + h _ 25J- 2 z^ -f- 2 ^3 _ 2 z^ + 2:, , 2 2:3 -f z^ ^^^ 6 6 6 2/1 = ^1 + -3- (^2 — ^1) = ^3 ^ as also 2/, = -^^3-^ hence there follows ^» ^ !,^ ^ ^^ = ^^ j ^^, and 3 2 V. = [^0 + 3 (y, + 2/2) + 2/3] -^5 as also Whilst the foregoing formula for y^ is applicable when the surfaceis resolved into an even number of strips, the latter is employedwhen the number of these portions is an odd one. Consequently, we may also put approximately: ydx =j if {x) dx = [2/0 -1- 3 (y, + 2/2) + 2/3] ~^^if are four determined values of the function y = (f {x). 64 ELEMENTS OF ANALYSIS. [Art. 38. (vid. example, Art. 30), we have c = 1, Cj = 2, and 0 (x) = -; hence, there follows and the approximate value of this integral: J^2Q^ 111 .— = [l + 3(H-|) + i]. 1=^ = 0,694.
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