. An elementary treatise on the differential and integral calculus. m = 2r + 1) ;01 = y\an™ 6 (1 + tan2 (9)-^ tan 0, (when m -\- n = — 2r); /n—lx™ (i _ x2) 2 ^ (wneu x — sjn (9)# (157) 51. I xn cos a# ofa; ,Jcfl~\ • . \ w(w—i) r n 2 . /1Kft. = —g-ta sm <z# + n cos #z) *——z / zn~2 cos ## c/z.(159) oax cos«-i # (# cos # + ^ sin #) J)- (162) 54. * a + Z> cos 0 2 ^/a2—b2 LV< + 1 rvH«+V^-fltann = , log —= -0 , (when «<£). VJ8-o2 &Lv^ + a_A/^-atan|J /» 7* /y2 ^X\ Zn+1 * v; 1-2-3 . ..^i Wz/1-2-3 ... (w + 1)w)(,H^)+et°- (165) CHAPTER VI. LENGTHS OF CURVES, 171. Length of Plan
. An elementary treatise on the differential and integral calculus. m = 2r + 1) ;01 = y\an™ 6 (1 + tan2 (9)-^ tan 0, (when m -\- n = — 2r); /n—lx™ (i _ x2) 2 ^ (wneu x — sjn (9)# (157) 51. I xn cos a# ofa; ,Jcfl~\ • . \ w(w—i) r n 2 . /1Kft. = —g-ta sm <z# + n cos #z) *——z / zn~2 cos ## c/z.(159) oax cos«-i # (# cos # + ^ sin #) J)- (162) 54. * a + Z> cos 0 2 ^/a2—b2 LV< + 1 rvH«+V^-fltann = , log —= -0 , (when «<£). VJ8-o2 &Lv^ + a_A/^-atan|J /» 7* /y2 ^X\ Zn+1 * v; 1-2-3 . ..^i Wz/1-2-3 ... (w + 1)w)(,H^)+et°- (165) CHAPTER VI. LENGTHS OF CURVES, 171. Length of Plane Curves referred to Rectan-gular Axes.—Let P and Q be two consecutive points onthe curve AB, and let (x, y) be thepoint P ; let s denote the length ofthe curve AP measured from a fixedpoint A up to P. Then PQ = ds, PR == dx, EQ = dy, Therefore, from the right-angledtriangle PRQ we have = Vdz. hence, s = / y/dx* + dy2 = / ( i + cltYdx 1 + dxV m To apply this formula to any particular curve, we find dtithe value of -j- in terms of x from the equation of the curve, and then by integration between proper limits sbecomes known. The process of finding the length of an arc of a curve iscalled the rectification of the curve. It is evident that if y be considered the independentvariable, we shall have J dy. -A i + chfl The curves whose lengths can be obtained in finite termsare very limited in number. We proceed to consider someof the simplest applications; 348 RECTIFICATION OF TEE PARABOLA. 172. The Parabola.—The equation of the parabola is y2 = %px; hence, -^- = -• ax y ... .=/(i+J)**; * = *, or s =-f(p*+ y2)idg, (which, by Ex. 35. Art. 146) = yA/^+y3 +1 log (y + VW+f) + a (i) If we estimate the arc from the vertex, then s = 0,y = 0, and we have 0=§logi>+tf; .-. 0=-£]ogp, which in (1) gives which is the length of the curve from the vertex to thepoint which has any ordina
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