Elements of analysis as applied to the mechanics of engineering and machinery . the cor-responding ordinate of the point ^^ !/ = ?/, +2/. = 1—^-1;further, since we have for a; ^ 2,y^ = i and y^=—f, there followsalso the co-ordinate of the point31: y = 4: — f = |. Likewise,we obtain for x = 3, y = y^ -j-y^ = d — 9 = 0,forx = i, y =16 — 6_4 ^ _ 1^% for ^ = _ 1, 2/ = 1 + ^ = I, for ^ = _ 2,2/ = 4 -f- I = 2_o^ ^(3., and we perceive that the last curve has,from A to the right side,the courseAWIIKL,..^ at the commence-ment of which it passes alongabove the abscissa AK=S^ butthat after the point K it
Elements of analysis as applied to the mechanics of engineering and machinery . the cor-responding ordinate of the point ^^ !/ = ?/, +2/. = 1—^-1;further, since we have for a; ^ 2,y^ = i and y^=—f, there followsalso the co-ordinate of the point31: y = 4: — f = |. Likewise,we obtain for x = 3, y = y^ -j-y^ = d — 9 = 0,forx = i, y =16 — 6_4 ^ _ 1^% for ^ = _ 1, 2/ = 1 + ^ = I, for ^ = _ 2,2/ = 4 -f- I = 2_o^ ^(3., and we perceive that the last curve has,from A to the right side,the courseAWIIKL,..^ at the commence-ment of which it passes alongabove the abscissa AK=S^ butthat after the point K it runs be-low XX indefinitely, whilst tothe left of A, it ascends constant-ly, forming the indefinite branchAF , From the above also,FT is a point of inflection, and Ma maximum point, of the at A and Jf the curve hasthe direction of JTX, at W itascends at an angle a = 45*^, be-cause we have for the same ^ = X (2 — x) = 1; but for theangle of inclination at K^ thereis tang, a = — 3, consequently,a = 11 34^ &c. The quadrature of the curve. _X X X is performed by the integralF= fydx = C(x^ — x^ X* ^^ r 1 ^ T ~~ 12 TV ~ , there follows, for example, for the portion of surface ) dx= Cxdx^ljxd )• [Art. 35. ELEMENTS OF ANALYSIS. 55 33A WMKabove AK=^, the area i^= — (1 _ |) = |, and, on the other hand, for the portion 3X4 below the portion 34 of the abscissa,i^, = y(l-|)-|-(l-|) = 0_f = -|.Lastly, to find the length of a portion of a curye, as A W2I^ we put s =J V 1 -^ x (2 — xf dx = P <p (x) dx, and apply the method of integration discussed in Art. 30. We havehere c ^ 0 and c^ = 2] if we assume n = 4, there follows dx = ^^^^ = ?^^ ^ 1, and if in the function c (x) = Vl-{-x(2 — xY,n 4 we substitute for x the values 0, i, 1, f, 2, successively, there will result: ^ (0) =: 1/T = 1, ^ Q) = i/iT:S = h cp (1) = v^l-l-l =.: 1/2 = 1,414 . . ^ (I) = lT+TV = I, and c. (2) =.1/1 = 1,and hence the length of the arc A WM: s = [<P (0)
Size: 923px × 2708px
Photo credit: © The Reading Room / Alamy / Afripics
License: Licensed
Model Released: No
Keywords: ., bookcentury1800, bookdecade1860, bookpublisherphiladelphiakingba
