. Differential and integral calculus, an introductory course for colleges and engineering schools. e now use s to denote the arc AB, we have (1) 8= f Vl +y2dx, or s = f Vdx2 + dy2. J a J a The second form of the formula is the more convenient when thecurve is given by parametric equations. When the curve is given in polar coordinates, we have, byformulae V and V of Art. 102, D9s = Vp2+ (Dep)2, and ds = Vp2 + p2 dd,whence Vp2 + process of finding the length of a curve is termed rectification. Example 1. Let us find the length of an arc of the upper branch ofthe semicubical parabola 9 i
. Differential and integral calculus, an introductory course for colleges and engineering schools. e now use s to denote the arc AB, we have (1) 8= f Vl +y2dx, or s = f Vdx2 + dy2. J a J a The second form of the formula is the more convenient when thecurve is given by parametric equations. When the curve is given in polar coordinates, we have, byformulae V and V of Art. 102, D9s = Vp2+ (Dep)2, and ds = Vp2 + p2 dd,whence Vp2 + process of finding the length of a curve is termed rectification. Example 1. Let us find the length of an arc of the upper branch ofthe semicubical parabola 9 if = 4 xz, measured from the origin to the point whose abscissa is 3. From thegiven equation we have, for the upper branch, V = 5 x*, V = *K Vl + y2 = Vl +x. §167 LENGTHS OF CURVES 237 Hence, by formula (1) above, arc OP = fVl+zdx= \|~(1+a;)*Y= ^.Jo 3L Jo 3 The parametric equations of this curve may be had by writing y = txin the given equation. There results V ,9x = -t2, y t\ In order to obtain the length of the arc fromthese equations, we differentiate them and get 27 dx = — t dt, dy t2dt,. Vdx2 + dy2 = ^Vi + 9t2tdt. Moreover, when x = 0, t = 0, and when xHence by the second part of formula (1), 3, i = V3 arc OP = 7 CV~W± + 9t2tdt=-n Cv* V4 + 9 t2d(4: + 9*2) 4J0 8J0 = ^[(4 + 9#] N3 ^(64-8) =| Example 2. Let us find the length of one turn of the logarithmicspiral p = aebd from B = 0 to d = x. We have P = a&e6*, VP2 + P2 = a VI + b2e™.Then, by formula (2), s - a VT+lf- Cehe dd = j Vr^¥(eb°yo Jo 0 = |vr+iT2(^ -1). s is the heavy arc in the figure. When p = 0, 0 = — go , and from thiswe know that between any point of theplane, as A, and the pole there are aninfinite number of turns of the curve (theL part of the curve for which 0 is negativeThe figure is drawn for b=VA^ayd^o is dotted in the figure). We can now show very easily that the total length of
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